
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 $f(x) = \cos^2 \left(x + \frac{\pi}{12}\right)$, $g(x) = 1 + \frac{1}{2}\sin 2x$.  
(1) Suppose $x = x_0$ is the axis of symmetry for the graph of the function $y = f(x)$, find the value of $g(x_0)$.  
(2) Let the function $h(x) = f(x) + g(x)$. If the inequality $|h(x) - m| \leq 1$ holds for all $x$ in the interval $\left[-\frac{\pi}{12}, \frac{5\pi}{12}\right]$, determine the range of values for the real number $m$.

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