
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
Let  $a_1,\ldots,a_8$  be reals, not all equal to zero. Let 
\[ c_n = \sum^8_{k=1} a^n_k\] 
for  $n=1,2,3,\ldots$ . Given that among the numbers of the sequence  $(c_n)$ , there are infinitely many equal to zero, determine all the values of  $n$  for which  $c_n = 0.$ 

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