
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
If \( x, y \in [0, \pi] \), and for any real number \( t \), the expression \(\left(-2 \cos t - \frac{1}{2} \cos x \cos y\right) \cos x \cos y - 1 - \cos x + \cos y - \cos 2t\) is always negative. Find the relationship that \( x \) and \( y \) must satisfy.

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