
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
In trapezoid $ABCD$, where $\overline{AB}\parallel\overline{CD}$ and sides $BC=CD=50$, the diagonal $\overline{AC}$ and $\overline{BD}$ intersect at point $O$. Point $P$ is located on $\overline{BD}$ such that $BP:PD=1:1$. Given that $OP=13$ and $\overline{AD}\perp\overline{BD}$, determine the length of diagonal $AC$ in its simplest form, expressible as $m\sqrt{n}$, where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime. Calculate $m+n$.
**A)** 40  
**B)** 74  
**C)** 122  
**D)** 228

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