
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 \( AB = 13 \), \( BC = 14 \), and \( AC = 15 \). Point \( Y \) lies on the interior of \( \overline{BC} \), and points \( J_1 \) and \( J_2 \) are the incenters of \( \triangle ABY \) and \( \triangle ACY \), respectively. Find the minimum possible area of \( \triangle AJ_1J_2 \) as \( Y \) varies along \( \overline{BC} \).

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