
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
Arrange all simplest proper fractions with a denominator of 24 in ascending order, denoted as \(a_{1}, a_{2}, \cdots, a_{n}\). Determine the value of \(\sum_{i=1}^{n} \cos \left(a_{i} \pi\right)\).

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