
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
The corners of a  $2$ -dimensional room in the shape of an isosceles right triangle are labeled  $A$ ,  $B$ ,  $C$  where  $AB = BC$ . Walls  $BC$  and  $CA$  are mirrors. A laser is shot from  $A$ , hits off of each of the mirrors once and lands at a point  $X$  on  $AB$ . Let  $Y$  be the point where the laser hits off  $AC$ . If  $\tfrac{AB}{AX} = 64$ ,  $\tfrac{CA}{AY} = \tfrac pq$  for coprime positive integers  $p$ ,  $q$ . Find  $p + q$ .

*Proposed by Sid Doppalapudi*

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