
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
Let the function $f(x) = x^2 + 2ax - a - 1$, where $x \in [0,2]$ and $a$ is a constant.

(1) When $a=1$, find the range of $f(x)$;

(2) Find the expression for the minimum value $g(a)$ of $f(x)$; determine if there exists a smallest integer $m$ such that $g(a) - m \leqslant 0$ holds for any $a \in \mathbb{R}$. If it exists, find the value of $m$; if not, explain why.

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