
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 $f(x) = a^{x-a} + 1$, where $a > 0$ and $a \neq 1$, it always passes through the fixed point $(2, 2)$.
1. Find the real number $a$;
2. Under the condition of (1), the graph of function $f(x)$ is translated 1 unit downward and then $a$ units to the left to obtain function $g(x)$. Let the inverse function of $g(x)$ be $h(x)$, find the expression for $h(x)$;
3. For the function $y = h(x)$ defined on the interval $(1, 4]$, if in its domain, the inequality $\left[h(x) + 2\right]^2 \leq h(x^2) + h(x)m + 6$ always holds, find the range of values for $m$.

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