
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 $f(x) = ax^2 + bx + c$ whose graph passes through the point $(-1,0)$, is there a set of constants $a$, $b$, and $c$ that satisfies the inequality $x \leqslant f(x) \leqslant \frac{1}{2}(1 + x^2)$ for all real numbers $x$? If such constants exist, find the values of $a$, $b$, and $c$; if not, explain why.

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