
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 a function $f(x) = ax^2 + bx + c \ (c > 0)$ that is even, the tangent line to the graph of $y = f(x)$ at the point $(1, f(1))$ is parallel to the line $2x - y - 3 = 0$. Let another function $g(x) = \frac{e^x}{f(x)}$.  
(1) Find the values of $a$ and $b$.  
(2) Discuss the monotonicity of $g(x)$.  
(3) If $x_0$ is a point of local minimum for $g(x)$, find the range of $g(x_0)$.

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