
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
Calculate \( t(0) - t(\pi / 5) + t(2 \pi / 5) - t(3 \pi / 5) + \ldots + t(8 \pi / 5) - t(9 \pi / 5) \), where \( t(x) = \cos 5x + *\cos 4x + *\cos 3x + *\cos 2x + \ldots \). An analysis shows that the coefficients marked with * are unknown. Can this sum be calculated without knowing the values of the erased coefficients?

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