
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  $ABC$  be a triangle and  $\Gamma$  the  $A$ - exscribed circle whose center is  $J$  . Let  $D$  and  $E$  be the touchpoints of  $\Gamma$   with the lines  $AB$  and  $AC$ , respectively. Let  $S$  be the area of the quadrilateral  $ADJE$ , Find the maximum value that  $\frac{S}{AJ^2}$  has and when equality holds.

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