
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 trapezium in which  $AB //CD$  and  $AD \perp AB$ . Suppose  $ABCD$  has an incircle which touches  $AB$  at  $Q$  and  $CD$  at  $P$ . Given that  $PC = 36$  and  $QB = 49$ , find  $PQ$ .

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