
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
Let the function \( f(x) \) have a derivative \( f'(x) \) on \( \mathbf{R} \), such that for any \( x \in \mathbf{R} \), \( f(x) + f(-x) = x^2 \). In the interval \( (0, +\infty) \), \( f'(x) > x \). If \( f(1+a) - f(1-a) \geq 2a \), then the range of the real number \( a \) is ______.

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