
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]


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Given the quadratic function \( f(x) \) satisfying the following conditions: (1) \( f(-1) = 0 \); (2) \( x \leq f(x) \leq \frac{x^2 + 1}{2} \), and \( x \in \mathbb{R} \). Find the explicit form of this quadratic function.

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