
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 $f(x) = (a + 2\cos^2 \frac{x}{2})\cos(x + \frac{\pi}{2})$, and $f(\frac{\pi}{2}) = 0$.

(Ⅰ) Find the value of the real number $a$;  
(Ⅱ) If $f(\frac{\alpha}{2}) = -\frac{2}{5}$, where $\alpha \in (\frac{\pi}{2}, \pi)$, find the value of $\cos(\frac{\pi}{6} - 2\alpha)$.

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