
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 the quadratic function $f(x)=-x^{2}+ax+b (a,b \in \mathbb{R})$, let $M(a,b)$ be the maximum value of the function $g(x)=|f(x)|$ on the interval $[1,2]$.
(1) When $a=1$, find the expression for $M(1,b)$ in terms of $b$;
(2) If for any $a$ and $b \in \mathbb{R}$, it is always true that $M(a,b) \geq M(a_{0},b_{0})$, find all real number pairs $(a_{0}, b_{0})$ that satisfy this condition.

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