
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
[Rectangle](https://artofproblemsolving.com/wiki/index.php/Rectangle) $ABCD$ is given with $AB=63$ and $BC=448.$ Points $E$ and $F$ lie on $AD$ and $BC$ respectively, such that $AE=CF=84.$ The [inscribed circle](https://artofproblemsolving.com/wiki/index.php/Inscribed_circle) of [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) $BEF$ is [tangent](https://artofproblemsolving.com/wiki/index.php/Tangent) to $EF$ at point $P,$ and the inscribed circle of triangle $DEF$ is tangent to $EF$ at [point](https://artofproblemsolving.com/wiki/index.php/Point) $Q.$ Find $PQ.$

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