
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 $f(x)$ defined on $\mathbb{R}$ satisfy $f(-x) + f(x) = x^2$. If $f'(x) < x$ for $x < 0$, then the solution set for the inequality $f(x) + \frac{1}{2} \geq f(1-x) + x$ is _________.

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