
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 with  $\angle C = 90^o$  and  $AC = 1$ . The median  $AM$  intersects the incircle at the points  $P$  and  $Q$ , with  $P$  between  $A$  and  $Q$ , such that  $AP = QM$ . Find the length of  $PQ$ .

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