
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  $\{ z_n \}_{n \ge 1}$  be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer  $k$ ,  $|z_k z_{k+1}|=2^k$ . Denote  $f_n=|z_1+z_2+\cdots+z_n|,$  for  $n=1,2,\cdots$ (1) Find the minimum of  $f_{2020}$ .
(2) Find the minimum of  $f_{2020} \cdot f_{2021}$ .

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