
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 $a\geqslant 3$, and the function $F(x)=\min\{2|x-1|,x^{2}-2ax+4a-2\}$, where $\min\{p,q\}= \begin{cases} p, & p\leqslant q \\ q, & p > q \end{cases}$.  
$(1)$ Find the minimum value $m(a)$ of $F(x)$;  
$(2)$ Find the range of $x$ for which the equation $F(x)=x^{2}-2ax+4a-2$ holds.

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