
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:

\[
\left(\sqrt{2(1+\cos 2x)} - \sqrt{9-\sqrt{7}} \sin x + 1\right) \cdot \left(3 + 2 \sqrt{13 - \sqrt{7}} \cos y - \cos 2y\right)
\]

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

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