
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 $f(x)$ defined on the domain $D$, if there exists a constant $M > 0$ such that for any $x \in D$, $|f(x)| \leqslant M$ holds, then $f(x)$ is called a bounded function on $D$. The constant $M$ is referred to as the upper bound of the function $f(x)$. Consider the function $f(x) = 2 + a\sin x - \cos^{2} x$.

(1) When $a = -2$, find the range of the function $f(x)$ and determine whether $f(x)$ is a bounded function for any $x \in \mathbb{R}$. Please provide a reasoning for your answer.

(2) If the function $f(x)$ is bounded by 4 for any $x \in \mathbb{R}$, find the range of possible values for the real number $a$.

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