
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
Given real numbers  $b_0,b_1,\ldots, b_{2019}$  with  $b_{2019}\neq 0$ , let  $z_1,z_2,\ldots, z_{2019}$  be the roots in the complex plane of the polynomial
\[
P(z) = \sum_{k=0}^{2019}b_kz^k.
\]
Let  $\mu = (|z_1|+ \cdots + |z_{2019}|)/2019$  be the average of the distances from  $z_1,z_2,\ldots, z_{2019}$  to the origin.  Determine the largest constant  $M$  such that  $\mu\geq M$  for all choices of  $b_0,b_1,\ldots, b_{2019}$  that satisfy
\[
1\leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019.
\]

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