
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 $f(x) = e^x - ax^2 - bx - 1$, where $a, b \in \mathbb{R}$, and $e$ is the base of the natural logarithm. The inequality $|x-a| \geq f(x)$ always holds. Find the range of real numbers for $a$.  
(1) If the tangent line to the function $f(x)$ at the point $(1, f(1))$ is given by the equation $y = (e-1)x - 1$, find the values of the real numbers $a$ and $b$.  
(2) Let $g(x)$ be the derivative of the function $f(x)$, find the minimum value of $g(x)$ in the interval $[0, 1]$.

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