
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
Determine the number of  $8$ -tuples  $(\epsilon_1, \epsilon_2,...,\epsilon_8)$  such that  $\epsilon_1, \epsilon_2, ..., 8 \in \{1,-1\}$  and  $\epsilon_1 + 2\epsilon_2 + 3\epsilon_3 +...+ 8\epsilon_8$  is a multiple of  $3$ .

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