
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 function  $f(x)=x^{2}+ax+b$  with  $a,b \in R$ , if there exists a real number  $m$  such that  $\left| f(m) \right| \leq \frac{1}{4}$  and  $\left| f(m+1) \right| \leq \frac{1}{4}$ , then find the maximum and minimum of the value of  $\Delta=a^{2}-4b$ .

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