
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 $f(x) = x^2 + 4x + 3$ ($x \in \mathbb{R}$), and the graph of $g(x)$ is symmetric to the graph of $f(x)$ about the line $x=1$.
(1) Find $g(x)$;
(2) If the solution set of the inequality $g(x) \geq g(a) - 4$ for $x$ is the set of all real numbers, find the maximum value of $a$.

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