
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
Let $\bar{a}$ be a vector with coordinates $\bar{a}(3,-4,-\sqrt{11})$, and $\bar{b}$ be a vector with coordinates $\bar{b}(\sin 2x \cos y, \cos 2x \cos y, \sin y)$. The length $|\bar{a}|$ is 6, and the length of vector $\bar{b}$ is 1. Since $f(x, y)=\bar{a} \cdot \bar{b}$, where $\bar{a} \cdot \bar{b}$ is the dot product of the vectors, by the properties of the dot product $-|\bar{a}||\bar{b}| \leq \bar{a} \cdot \bar{b} \leq|\bar{a}||\bar{b}|$, we get $-6 \leq f(x, y) \leq 6$. This equality will be achieved when the vectors $\bar{a}$ and $\bar{b}$ are collinear, i.e., $\frac{\sin 2x \cos y}{3}=\frac{\cos 2x \cos y}{-4}=\frac{\sin y}{-\sqrt{11}}$ which is satisfied for appropriate $x$ and $y$. Thus, the minimum value of $f(x, y)$ is -6 and the maximum value is 6.

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