
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
Consider all triangles $ABC$ where $AB = AC$, $D$ is a point on $\overline{AC}$ such that $\overline{BD} \perp \overline{AC}$. Assume $AC$ and $CD$ are odd integers, and $BD^{2} = 65$. Find the smallest possible value of $AC$.

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