
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
An isosceles triangle with  $AB=AC$  has an inscribed circle  $O$ , which touches its sides  $BC,CA,AB$  at  $K,L,M$  respectively. The lines  $OL$  and  $KM$  intersect at  $N$ ; the lines  $BN$  and  $CA$  intersect at  $Q$ . Let  $P$  be the foot of the perpendicular from  $A$  on  $BQ$ . Suppose that  $BP=AP+2\cdot PQ$ . Then, what values can the ratio  $\frac{AB}{BC}$  assume?

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