
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 - √e)x - a, a function f(x) is defined on R and satisfies f(-x) + f(x) = x^2 and f'(x) < x when x ≤ 0. If there exists x₀ ∈ {x | f(x) + 1/2 ≥ f(1 - x) + x} such that x₀ is a zero point of the function y = g(x) - x, determine the range of the real number a.

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