
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  $z$  be an integer  $> 1$  and let  $M$  be the set of all numbers of the form  $z_k = 1+z + \cdots+ z^k, \  k = 0, 1,\ldots$ . Determine the set  $T$  of divisors of at least one of the numbers  $z_k$  from  $M.$ 

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