
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
The circumcenter of triangle $\triangle ABC$ is $O$, and the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, $c$, respectively. Given that $\overrightarrow{AO} \cdot \overrightarrow{BC} = \frac{1}{2}a\left(a-\frac{8}{5}c\right)$ and $b=4$. Find the maximum value of the area of $\triangle ABC$.

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