
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 quadratic function \( f(x) = ax^2 + bx + c \) where \( a \), \( b \), and \( c \) are real numbers and \( a \neq 0 \). The following conditions hold: (1) \( f(-1) = 0 \), (2) For any \( x \in \mathbb{R} \), \( x \leq f(x) \leq \frac{1}{2}(x^2 + 1) \). Find the value of \( a \).

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