
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
16. Given that \( a \) is a real constant, and the function
\[ 
f(x) = e^{-x} \sin x + ax \quad \text{for } x \in [0, 2\pi] 
\]

(1) Let \( g(x) \) be the derivative of \( f(x) \). Determine the intervals of monotonicity of \( g(x) \) on the interval \([0, 2\pi]\).

(2) If \( f(x) \) has exactly one local maximum and one local minimum in the interval \((0, 2\pi)\), find the range of values for the real number \( a \).

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