
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 \( f(x) = a \sin x - \frac{1}{2} \cos 2x + a - \frac{3}{a} + \frac{1}{2} \), where \( a \in \mathbb{R} \) and \( a \neq 0 \):
(1) If \( f(x) \leq 0 \) for all \( x \in \mathbb{R} \), find the range of \( a \).
(2) If \( a \geq 2 \) and there exists an \( x \in \mathbb{R} \) such that \( f(x) \leq 0 \), find the range of \( a \).

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