
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  $ABCD$  be a convex quadrilateral with  $AC \perp BD$ , and let  $P$  be the intersection of  $AC$  and  $BD$ . Suppose that the distance from  $P$  to  $AB$  is  $99$ , the distance from  $P$  to  $BC$  is  $63$ , and the distance from  $P$  to  $CD$  is  $77$ . What is the distance from  $P$  to  $AD$ ?

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