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Part of me was hoping broken morningstars lost their teeth or something. ¯\_(ツ)_/¯

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Tenibri retrieves the gem he found and throws it into the room, near Elessia. The gem cracks cleanly in two, the two halves sinking briefly into the softening earth as the ground begins rising into a vaguely humanoid form.
Move: retrieve elemental gem
Standard: activate elemental gem with a 45' throw
I'm basing the distance for the throw on the range of a summon nature's ally V spell (i.e. caster level 9); also, summon nature's ally is a 1 round casting time, so the elemental won't act until next round.

Oxnard Kettlebeak |

Round 1
As the skeletons move in, Seldiir blasts one of them to pieces with a well timed hit.
When he had a chance again, Seldiir swings, but the skeleton remains standing after this one.
Angrboda adds Faerie Fire to the Glitterdust and you might as well be full able to see Elessia now!
Eniac moves forward boldly!
Round 2
Tenibri throws the elemental stone and summons the earth elemental.
Turn Order:
1. Tenibi
2. Skeletons
3. Loam Yin
4. Elessia
5. Seldiir
6. Angrboda
7. Lara
8. Dax
Status:
Seldiir: 22/28
South skeleton: -16
Pending Lara in round 1

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Lucky Point Blank Precise Shot: 1d20 + 8 ⇒ (5) + 8 = 13
1d6 + 5 ⇒ (3) + 5 = 8
Lara nocks an arrow and launches it at the blinded, glittery, faerie-fire-glowy witch.

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Dax flies down the corridor.
fly: 1d20 + 10 ⇒ (8) + 10 = 18
And banks around the corner to get past Seldiir’s bulk.
The mephit then scowls at the unharmed skeleton and breathes fire at it.
fire damage: 1d8 ⇒ 7
dc13 reflex for half

Oxnard Kettlebeak |

Round 1
Lara misses.
Reflex: 1d20 + 4 ⇒ (9) + 4 = 13
Dax breathes, but the skeleton dodges some of it.
Round 2
The Skeletons swing their morning stars again:
Broken Morningstar: 1d20 + 5 ⇒ (6) + 5 = 11
Damage: 1d8 + 3 ⇒ (8) + 3 = 11
Broken Morningstar: 1d20 + 5 ⇒ (2) + 5 = 7
Damage: 1d8 + 3 ⇒ (2) + 3 = 5
both miss!
Elessia begins casting a spell!
Turn Order:
1. Tenibi
2. Skeletons
3. Loam Yin
4. Elessia
5. Seldiir
6. Angrboda
7. Lara
8. Dax
Status:
Seldiir: 22/28
South skeleton: -16
North Skeleton: -4
the party can act again!

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Dax races round into a flanking position and attacks the north skeleton.
Claws do bludgeoning and slashing damage.
attack, flanking: 1d20 + 5 + 2 ⇒ (18) + 5 + 2 = 25
damage: 1d3 + 1 ⇒ (3) + 1 = 4
attack, flanking: 1d20 + 5 + 2 ⇒ (13) + 5 + 2 = 20
damage: 1d3 + 1 ⇒ (3) + 1 = 4

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Spellcraft (identify spell), investigative mind: 1d20 + 14 ⇒ (14) + 14 = 28
Spellcraft (identify spell), investigative mind: 1d20 + 14 ⇒ (2) + 14 = 16
Tenibri's action will depend on the spell being cast.

Summoned Monster: Tenibri |

The earth elemental brings two full-bodied arms down on Elessia, but may be showing signs of needing to acclimate to a new plane.
Full round: full attack (10' reach)
Slam, power attack: 1d20 + 14 - 3 ⇒ (1) + 14 - 3 = 12
Damage, B, power attack: 2d4 + 7 + 6 ⇒ (2, 4) + 7 + 6 = 19
Slam, power attack: 1d20 + 14 - 3 ⇒ (7) + 14 - 3 = 18
Damage, B, power attack: 2d4 + 7 + 6 ⇒ (3, 2) + 7 + 6 = 18

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As the skeletons are being dealt with, Loam presses further into the room. From behind the safety of the elemental, she tries to disrupt Elessia's casting.
+2 Pistol vs. Touch: 1d20 + 8 ⇒ (16) + 8 = 24
Damage: 1d6 + 3 ⇒ (1) + 3 = 4

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Lucky Point Blank Precise Shot: 1d20 + 8 ⇒ (7) + 8 = 15
1d6 + 5 ⇒ (4) + 5 = 9
Nonplussed at her poorly-aimed shot, Lara tries again.

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Can he use the conjurer's focus exploit (at a cost of 3 arcane reservoir points) to power the counterspell exploit (at a cost of 1 arcane reservoir point)? The text of both exploits below:
By expending 1 point from her arcane reservoir, the arcanist can attempt to counter a spell as it is being cast. She must identify the spell being cast as normal. If she successfully does so, the arcanist can attempt to counter the spell as an immediate action and by expending an available arcanist spell slot of a level at least one higher than the level of the spell being cast. To counterspell, the arcanist must attempt a dispel check as if using dispel magic. If the spell being countered is one that the arcanist has prepared, she can instead expend an available arcanist spell slot of the same level, and she receives a +5 bonus on the dispel check. Counterspelling in this way does not trigger any feats or other abilities that normally occur when a spellcaster successfully counters a spell.
An occultist can spend 1 point from her arcane reservoir to cast summon monster I. She can cast this spell as a standard action and the summoned creatures remain for 1 minute per level (instead of 1 round per level). At 3rd level and every 2 levels thereafter, the power of this ability increases by one spell level, allowing her to summon more powerful creatures (to a maximum of summon monster IX at 17th level), at the cost of an additional point from her arcane spell reserve per spell level. An occultist cannot have more than one summon monster spell active in this way at one time. If this ability is used again, any existing summon monster immediately ends. This ability replaces the arcanist exploit gained at 1st level.

Oxnard Kettlebeak |

It turns out, she can't possibly make the concentration check against 18 damage from the Earth elemental so counterspelling isn't relevant. Otherwise it seems to me like it would work.
Round 2
Loam Yin hits with a bullet.
Seldiir smashes at the north skeleton, and when Dax hits it as well it crumbles.
Lara misses, her arrow deflected away by a protective magic.
Round 3
The earth elemental slams Elessia once, and her concentration falters losing her cast spell.
Turn Order:
1. Tenibi
2. Skeletons
3. Loam Yin
4. Elessia
5. Seldiir
6. Angrboda
7. Lara
8. Dax
Status:
Seldiir: 22/28
South skeleton: -16
North Skeleton: -12
Elessia: -22
Pending Angrboda and Tenibri before the bad guys can act.

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Tenibri moves into the room, slipping through the congestion at the entrance to find a better position to act from inside.
Double move: 40'

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Like a fountain out of some mad groundskeeper's nightmare, Eniac launches another arcing stream of vitriol at the "witch"...
spit: 1d20 + 2 ⇒ (20) + 2 = 22
THREAT!: 1d20 + 2 ⇒ (5) + 2 = 7
acid damage: 1d8 ⇒ 4
...with eerily mathematical grace, although its trajectory calculations over such a distance likely have room for improvement.

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Sidling and staggering out past the giant Seldiir (while still seeming to be carefully tracing some invisible path), Angrboda is relentless in her magical assault, calling forth an eerie, Dwarf-sized hovering vortex of heavy, black air, and dancing with ominous flashes - it settles over the obviously-outclassed 'witch' and erupts in a bolt of energy!
aggressive thundercloud, CL 5th, DC 17 Reflex save to negate: 3d6 ⇒ (5, 1, 5) = 11

Oxnard Kettlebeak |
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Sorry- the "north" skeleton crumbled I left it on the run down in error there

Oxnard Kettlebeak |

Round 2
Reflex: 1d20 + 3 ⇒ (16) + 3 = 19
Elessia avoids the thunder cloud, but gets spit on by the slug.
Round 3
Tenbri moves in closer.
The final skeleton continues swinging at Seldiir:
B Morningstar: 1d20 + 5 ⇒ (7) + 5 = 12
Damage: 1d8 + 3 ⇒ (8) + 3 = 11
Elessia realizes she needs to take more aggressive action steps forward and channels negative energy twice (using quick channel).
Negative Energy: 3d6 ⇒ (5, 1, 1) = 7
Negative Energy: 3d6 ⇒ (1, 3, 3) = 7
This hits the Earth Ele, Loam Yin, Eniac, and Dax but misses Angrboda/Seldiir. DC 16 will for half (two saves)
Turn Order:
1. Tenibi
2. Skeletons
3. Loam Yin
4. Elessia
5. Seldiir
6. Angrboda
7. Lara
8. Dax
Status:
Seldiir: 22/28
South skeleton: -16
Elessia: -26
That brings up the party again

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The vortex follows Elessia, and strikes again...
aggressive thundercloud, CL 5th, DC 17 Reflex save to negate: 3d6 ⇒ (4, 3, 5) = 12
...while Angrboda, spurred on - and seemingly strengthened??? - by the strange and disturbing noises now coming from Eniac, continues her magical siege from below, stomping a ley line and causing a visible green pulse to shoot through the floor from under Angrboda's feet to under Elessia's!
thunderstomp, CL 5th, first-ever actual use of my Soothsayer Boon!: 1d20 + 12 ⇒ (8) + 12 = 20
I am now shaken for 1 round.

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Lucky Point Blank Precise Shot: 1d20 + 8 ⇒ (16) + 8 = 24
1d6 + 5 ⇒ (5) + 5 = 10
Steadying her breathing, Lara launches another arrow at the witch.

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Will saves:
Loam 1: 1d20 + 6 ⇒ (16) + 6 = 22
Loam 2: 1d20 + 6 ⇒ (15) + 6 = 21
Gukom 1: 1d20 + 6 ⇒ (18) + 6 = 24
Gukom 2: 1d20 + 6 ⇒ (3) + 6 = 9
Loam weathers the worst of Elessia's onslaught and fires back, hoping to end her quickly.
+2 Pistol vs. Touch: 1d20 + 8 ⇒ (14) + 8 = 22
Damage: 1d6 + 3 ⇒ (4) + 3 = 7

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Arrgh, forgot my better half!
Will save #1: 1d20 + 1 ⇒ (8) + 1 = 9
Will save #2: 1d20 + 1 ⇒ (10) + 1 = 11
Eniac quite visibly shrivels under the baleful power, a horrible sight to see and hear...
spit, melee penalty vs skeleton: 1d20 - 2 ⇒ (3) - 2 = 1
...and despite briefly casting an eyestalk at the skeleton nearby before retracting it, simply backs into the corner to pitifully vomit on the floor.

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Injured, Dax howls in pain and scoots around to flank the remaining skeleton.
claw attack, flanking: 1d20 + 5 + 2 ⇒ (8) + 5 + 2 = 15
damage (slashing and bludgeoning): 1d3 + 1 ⇒ (1) + 1 = 2
claw attack, flanking: 1d20 + 5 + 2 ⇒ (5) + 5 + 2 = 12
damage (slashing and bludgeoning): 1d3 + 1 ⇒ (2) + 1 = 3

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Tenibri finishes moving in closer, then throws a snowball.
Move: 20'
Standard: cast snowball
Ranged touch, snowball, into melee: 1d20 + 5 - 4 ⇒ (9) + 5 - 4 = 10
Damage, cold: 5d6 ⇒ (1, 2, 4, 4, 6) = 17

Oxnard Kettlebeak |

Round 3
Reflex: 1d20 + 3 ⇒ (1) + 3 = 4
Seldiir Swings, and crushes the final skeleton.
Angrboda hits Elessia in the Thundercloud, then stomps and drops her to the floor.
Lara hits the witch with her arrow, and Tenibri with a snowball and she falls to the ground!
End of combat
As you search the room, you find inside the massive dead lypenaia tree in the room Tessarael's Book of Infinite Spells!
Removing the artifact carefully, the magic that has been generated by the blackened tree ceases completely, and the Arcane Engine rumbles to a stop.
You return to your employer without further incident and receive your pay!

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"Some people are drawn to nearby magic, and rightfully so," Loam says. "I'm glad we managed to keep all this intact. If I have my way, I'll be studying it for weeks."

Oxnard Kettlebeak |

Tying up loose ends-
With her now visible you can tell that Elessia was a Changeling Cleric of Calistria, so she may have known about the engine based on her fey/hag connections.
Also, it becomes obvious now that she was using Vos and the hobgoblins- probably with a hefty amount of charm person involved. Her true purpose all along was to get to the magic in the basement.