
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
Suppose the quadratic function $f(x)=ax^2+bx+c$ has a maximum value $M$ and a minimum value $m$ in the interval $[-2,2]$, and the set $A={x|f(x)=x}$.

(1) If $A={1,2}$ and $f(0)=2$, find the values of $M$ and $m$.
(2) If $A={2}$ and $a\geqslant 1$, let $g(a)=M+m$, find the minimum value of $g(a)$.

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