
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
A wall made of mirrors has the shape of  $\triangle ABC$ , where  $AB = 13$ ,  $BC = 16$ , and  $CA = 9$ . A laser positioned at point  $A$  is fired at the midpoint  $M$  of  $BC$ . The shot reflects about  $BC$  and then strikes point  $P$  on  $AB$ . If  $\tfrac{AM}{MP} = \tfrac{m}{n}$  for relatively prime positive integers  $m, n$ , compute  $100m+n$ .

*Proposed by Michael Tang*

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