
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  $ABC$  with incenter  $I$  satisfy  $AB = 10$ ,   $BC = 21$ , and   $CA = 17$ . Points  $D$  and E lie on side  $BC$  such that  $BD = 4$ ,  $DE = 6$ , and  $EC = 11$ . The circumcircles of triangles  $BIE$  and  $CID$  meet again at point  $P$ , and line  $IP$  meets the altitude from  $A$  to  $BC$  at  $X$ . Find  $(DX \cdot EX)^2$ .

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