
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 $a$ and $b$ are constants, and $a \neq 0$, $f(x)=ax^{2}+bx$, $f(2)=0$.
(I) If the equation $f(x)-x=0$ has a unique real root, find the analytic expression of the function $f(x)$;
(II) When $a=1$, find the maximum and minimum values of the function $f(x)$ on the interval $[-1,2]$;
(III) When $x\geqslant 2$, the inequality $f(x)\geqslant 2-a$ always holds, find the range of the real number $a$.

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