
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 a quadratic function $f(x)=ax^{2}+bx+c$ defined on the interval $[-2,2]$, with maximum and minimum values $M$ and $m$ respectively. Let 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}
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