
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 $f(x) = (ax^2 + bx + c)e^{-x}$, its graph passes through the point $(0, 2a)$ and the slope of the tangent line at this point is $\frac{\pi}{4}$.  
(1) Express $b$ and $c$ in terms of $a$;  
(2) If $f(x)$ is not monotonic on the interval $\left[\frac{1}{2}, +\infty\right)$, find the range of values for $a$.

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