
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
Let triangle  $ABC$  have side lengths  $AB = 13$ ,  $BC = 14$ ,  $AC = 15$ . Let  $I$  be the incenter of  $ABC$ . The circle centered at  $A$  of radius  $AI$  intersects the circumcircle of  $ABC$  at  $H$  and  $J$ . Let  $L$  be a point that lies on both the incircle of  $ABC$  and line  $HJ$ . If the minimal possible value of  $AL$  is  $\sqrt{n}$ , where  $n \in \mathbb{Z}$ , find  $n$ .

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