
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
Two points $C$ and $D$ are on the same side of line $AB$, such that $\triangle ABC$ and $\triangle BAD$ are on a plane with $AB = 13$, $BC = 12$, $CA = 5$, $AD = 5$, and $DB = 12$. Furthermore, the angle $\angle BAC = \angle BAD$. The intersection of these two triangular regions has area $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

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