
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 function $f(x)=ax^{2}+bx+c$ where $a,b,c \in \mathbb{R}$ and $a \neq 0$, if the inequality $2x \leq f(x) \leq \frac{1}{2}(x+1)^{2}$ holds for any real number $x$.

1. Find the value of $f(1)$;
2. Find the range of values for $a$;
3. If the function $g(x)=f(x)+2a|x-1|$, $x\in[-2,2]$ has a minimum value of $-1$, find the value of $a$.

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