
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 $g(x) = e^x + (1 - \sqrt{e})x - a$, a continuous function $f(x)$ defined on $\mathbb{R}$ satisfies $f(-x) + f(x) = x^2$, and when $x \leq 0$, $f'(x) < x$. If there exists $x_0 \in \{x | f(x) + \frac{1}{2} \leq f(1 - x) + x\}$, and $x_0$ is a fixed point of the function $g(x)$, determine the range of the real number $a$.

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