
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 \(\triangle ABC\),
$$
\begin{array}{l}
z = \frac{\sqrt{65}}{5} \sin \frac{A+B}{2} + \mathrm{i} \cos \frac{A-B}{2}, \\
|z| = \frac{3 \sqrt{5}}{5}.
\end{array}
$$

Find the maximum value of \(\angle C\).

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