
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
Given the function $f(x) = a + 2\cos^2(x)\cos(2x + \theta)$ is an odd function, and $f\left(\frac{\pi}{4}\right) = 0$, where $a \in \mathbb{R}$, $\theta \in (0,\pi)$.

(1) Find the values of $a$ and $\theta$.

(2) If $f\left(\frac{\alpha}{4}\right) = -\frac{2}{5}$, $\alpha \in \left(\frac{\pi}{2},\pi\right)$, find the value of $\sin\left(\alpha + \frac{\pi}{3}\right)$.

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