
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
Consider the acute angle  $ABC$ . On the half-line  $BC$  we consider the distinct points  $P$  and  $Q$  whose projections onto the line  $AB$  are the points  $M$  and  $N$ . Knowing that  $AP=AQ$  and  $AM^2-AN^2=BN^2-BM^2$ , find the angle  $ABC$ .

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