
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
Find the smallest positive integer $k$ such that 
\[
z^{10} + z^9 + z^8 +z^6 +z^5 +z^4 +z+1
\]
divides \[
z^k-1.
\]

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