
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 \( t(0) - t(\pi / 5) + t\left((\pi / 5) - t(3 \pi / 5) + \ldots + t\left(\frac{8 \pi}{5}\right) - t(9 \pi / 5) \right) \), where \( t(x) = \cos 5x + * \cos 4x + * \cos 3x + * \cos 2x + * \cos x + * \). A math student mentioned that he could compute this sum without knowing the coefficients (denoted by *). Is he correct?

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