
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 triangle $\triangle ABC$, let $a$, $b$, and $c$ denote the lengths of the sides opposite angles $A$, $B$, and $C$ respectively, with $a=7$. Point $D$ is the midpoint of side $AC$. Given that $\overrightarrow{BD} \cdot \overrightarrow{AC} = \frac{25}{2}$, find the value of $\cos C$ when angle $C$ reaches its maximum value.

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