
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 $x_1$, $x_2$ be the two distinct roots of the function $f(x)=ax^2+(b-1)x+1$ ($a,b \in \mathbb{R}$, $a>0$).
(I) If $x_1=1$, and for any $x \in \mathbb{R}$, $f(2-x)=f(2+x)$, find $f(x)$;
(II) If $b=2a-3$, does the equation $f(x)=|2x-a|+2$ have negative real roots? If so, find the range of these negative roots. If not, explain why;
(III) If $a \geq 2$, $x_2-x_1=2$, and when $x \in (x_1, x_2)$, the maximum value of $g(x)=-f(x)+2(x_2-x)$ is $h(a)$, find the minimum value of $h(a)$.

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