
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 the graph of the quadratic function $f(x)$ passes through the point $(1,0)$, for any $x\in R$, $f(x)\leqslant 1$ always holds, and $f(2-x)=f(2+x)$:

(1) Find the analytical expression of $f(x)$.
(2) Let $g(x)=f(x)+ax$, where $x\in[-1,2]$, and the maximum value of $g(x)$ is $h(a)$, find $h(a)$.

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