
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 vectors $a = (\cos \frac{3}{2}x, \sin \frac{3}{2}x)$ and $b = (-\sin \frac{x}{2}, -\cos \frac{x}{2})$, where $x \in [\frac{\pi}{2}, \pi]$,
(1) If $|a + b| = \sqrt{3}$, find the value of x;
(2) For the function $f(x) = \vec{a} \cdot \vec{b} + |\vec{a} + \vec{b}|^2$, if $c > f(x)$ always holds, find the range of the real number c.

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