
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  $AB = 3$ ,  $AC = 4$ , and  $BC = 5$ . Suppose that  $D$  and  $E$  lie on  $AB$  and  $AC$ , respectively, such that  $D$ ,  $I$ , and  $E$  are collinear and  $DE \perp AI$ . Points  $P$  and  $Q$  lie on side  $BC$  such that  $IP = BP$  and  $IQ = CQ$ , and lines  $DP$  and  $EQ$  meet at  $S$ . Compute  $SI^2$ .

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