
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
A student discovered in a research study that the values of the following four expressions are equal to the same constant:  
(1) $\cos(-60^\circ) + \cos(60^\circ) + \cos(180^\circ);$  
(2) $\cos(-27^\circ) + \cos(107^\circ) + \cos(227^\circ);$  
(3) $\cos(30^\circ) + \cos(150^\circ) + \cos(270^\circ);$  
(4) $\cos(40^\circ) + \cos(160^\circ) + \cos(280^\circ).$  
(I) Please choose one of the above expressions, simplify it, and find its value;  
(II) Based on the result of (I), write a general proposition that includes the four expressions given in the problem, and provide a proof.

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