
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 $f(x) = ax - a + 1$, where $(a > 0 \text{ and } a \neq 1)$ always passes through the fixed point $(3,2)$,
1. Find the real number $a$.
2. Under the condition of (1), after translating the graph of function $f(x)$ downward by 1 unit and then leftward by $a$ units to get function $g(x)$, let the inverse function of $g(x)$ be $h(x)$. Find the analytical expression of $h(x)$.
3. For the function $y = h(x)$ defined on the interval $[1,9]$, if within its domain, the inequality $(h(x) + 2)^2 \leq h(x^2) + m + 2$ always holds, find the range of values for $m$.

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