
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 parallelogram $ABCD$, $\angle{BAD}\cong\angle{BCD}$ and $AB = 10$, $BC = 12$, and $BD = 15$. Calculate the length of $CD$, which can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the value of $m + n$.

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