
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 that $a\geqslant 3$, the function $F(x)=\min\{2|x-1|,x^{2}-2ax+4a-2\}$, where $\min(p,q)= \begin{cases} p,p\leqslant q \\ q,p > q\end{cases}$
(I) Find the range of $x$ for which the equation $F(x)=x^{2}-2ax+4a-2$ holds.
(II) (i) Find the minimum value of $F(x)$, denoted as $m(a)$.
(ii) Find the maximum value of $F(x)$ on $[0,6]$, denoted as $M(a)$.

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