
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 a sequence of positive integers  $a_1, a_2, a_3, \dots, a_{n}$ , define the \emph{power tower function} \[f(a_1, a_2, a_3, \dots, a_{n})=a_1^{a_2^{a_3^{\mathstrut^{ .^{.^{.^{a_{n}}}}}}}}.\] Let  $b_1, b_2, b_3, \dots, b_{2017}$  be positive integers such that for any  $i$  between 1 and 2017 inclusive, \[f(a_1, a_2, a_3, \dots, a_i, \dots, a_{2017})\equiv f(a_1, a_2, a_3, \dots, a_i+b_i, \dots, a_{2017}) \pmod{2017}\] for all sequences  $a_1, a_2, a_3, \dots, a_{2017}$  of positive integers greater than 2017. Find the smallest possible value of  $b_1+b_2+b_3+\dots+b_{2017}$ .

[i]Proposed by Yannick Yao

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