
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  $ABCD$  be a square, and let  $M$  be the midpoint of side  $BC$ . Points  $P$  and  $Q$  lie on segment  $AM$  such that  $\angle BPD=\angle BQD=135^\circ$ . Given that  $AP<AQ$ , compute  $\tfrac{AQ}{AP}$ .

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