
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 functions $f(x)=-x^{3}+x^{2}+x+a$ and $g(x)=2a-x^{3}$ $(x\in\mathbb{R},a\in\mathbb{R})$.

$(1)$ Find the intervals of monotonicity for the function $f(x)$.

$(2)$ Find the extreme values of the function $f(x)$.

$(3)$ If for any $x\in[0,1]$, the inequality $g(x)\geqslant f(x)$ always holds, find the range of values for $a$.

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