
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In triangle $ABC$, $AC = 360$ and $BC = 240$. Points $H$ and $J$ are located on $\overline{AC}$ and $\overline{AB}$, respectively, such that $AH = HC$, and $\overline{CJ}$ is the angle bisector of $\angle C$. Let $Q$ be the point of intersection of $\overline{BH}$ and $\overline{CJ}$, and let $N$ be the point on line $BH$ such that $H$ is the midpoint of $\overline{QN}$. If $AN = 144$, find $JQ$.

Present the answer in LaTex format: \boxed{Your answer}
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