
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  $ABC$  we have  $AB=36$ ,  $BC=48$ ,  $CA=60$ . The incircle of  $ABC$  is centered at  $I$  and touches  $AB$ ,  $AC$ ,  $BC$  at  $M$ ,  $N$ ,  $D$ , respectively.  Ray  $AI$  meets  $BC$  at  $K$ . The radical axis of the circumcircles of triangles  $MAN$  and  $KID$  intersects lines  $AB$  and  $AC$  at  $L_1$  and  $L_2$ , respectively. If  $L_1L_2 = x$ , compute  $x^2$ .

*Proposed by Evan Chen*

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