
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 incenter  $I$ . The line through  $I$ , perpendicular to  $AI$ , intersects the circumcircle of  $ABC$  at points  $P$  and  $Q$ . It turns out there exists a point  $T$  on the side  $BC$  such that  $AB + BT = AC + CT$  and  $AT^2 =  AB \cdot AC$ . Determine all possible values of the ratio  $IP/IQ$ .

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