
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

Find the minimum value of the expression

\[
(3 \sqrt{2(1+\cos 2 x)}-\sqrt{8-4 \sqrt{3}} \sin x+2) \cdot(3+2 \sqrt{11-\sqrt{3}} \cos y-\cos 2 y)
\]

If the answer is not an integer, round it to the nearest whole number.

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