
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
$\triangle DEF$ is inscribed inside $\triangle ABC$ such that $D, E, F$ lie on $BC, AC, AB$, respectively. The circumcircles of $\triangle DEC, \triangle BFD, \triangle AFE$ have centers $O_1, O_2, O_3$, respectively. Given $AB = 15, BC = 20, AC = 17$, and $\stackrel{\frown}{BF} = \stackrel{\frown}{EC},\ \stackrel{\frown}{AF} = \stackrel{\frown}{CD},\ \stackrel{\frown}{AE} = \stackrel{\frown}{BD}$. Find the length of $BD$ and express it as a fraction $\frac{m}{n}$ where $m$ and $n$ are coprime integers. Then, find $m+n$.

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