
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 a function $f(x) = 2\sqrt{3}\sin\left(\frac{\pi}{4} + \frac{x}{2}\right)\sin\left(\frac{\pi}{4} - \frac{x}{2}\right) - \sin(\pi+x)$, and the graph of the function $y=g(x)$ is symmetric to the graph of $y=f(x)$ with respect to the line $x=\frac{\pi}{4}$.
(1) If there exists $x \in \left[0, \frac{\pi}{2}\right)$ such that the equation $g(x)^2 - mg(x) + 2 = 0$ holds, find the maximum and minimum values of the real number $m$.
(2) If the inequality $f(x) + ag(-x) > 0$ always holds when $x \in \left[0, \frac{11\pi}{12}\right]$, determine the range of values for $a$.

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