
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 $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $2c\cos B = 2a + b$.
(1) Find the measure of angle $C$;
(2) If the function $f(x) = 2\sin(2x + \frac{\pi}{6}) + m\cos 2x$ ($m \in \mathbb{R}$) has a symmetry axis with equation $x = \frac{C}{2}$ and $f(\frac{\alpha}{2}) = \frac{6}{5}$, find the value of $\cos(2\alpha + C)$.

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