
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 $f(x)=x|x-a|$, where $a\in R$, and $g(x)=x^{2}-1$.
(1) Solve the inequality $f(x)\geqslant g(x)$ when $a=1$;
(2) Denote the maximum value of the function $f(x)$ on the interval $[0,2]$ as $F(a)$, find the expression of $F(a)$.

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