
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 trapezoid $ABCD$ has sides $AB=104$, $BC=68$, $CD=23$, and $AD=94$, with $AB$ parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Determine the distance $AP$ if $AP= \frac{m}{n}$ with $m$ and $n$ being relatively prime integers.

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