
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
Trapezoid $ABCD$ has $\overline{AB}\parallel\overline{CD}$, and $\overline{AD}\perp\overline{BD}$. Both $BC$ and $CD$ have a length of $50$. $O$ is the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and $P$ is a point on $\overline{BD}$ such that $DP = \frac{2}{3}BD$. Given that $OP=15$, find the length of $AD$. Express $AD$ in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime.
A) $5\sqrt{2180}$
B) $5\sqrt{2183}$
C) $5\sqrt{2190}$
D) $5\sqrt{2200}$

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