
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
There exist complex numbers  $z_1,z_2,\dots,z_{10}$  which satisfy  $$ |z_ki^k+ z_{k+1}i^{k+1}| = |z_{k+1}i^k+ z_ki^{k+1}| $$  for all integers  $1 \leq k \leq 9$ , where  $i = \sqrt{-1}$ . If  $|z_1|=9$ ,  $|z_2|=29$ , and for all integers  $3 \leq n \leq 10$ ,  $|z_n|=|z_{n-1} + z_{n-2}|$ , find the minimum value of  $|z_1|+|z_2|+\cdots+|z_{10}|$ .

*Proposed by DeToasty3*

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