
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
Define a sequence  $\{a_n\}_{n \geq 1}$  recursively by  $a_1=1$ ,  $a_2=2$ , and for all integers  $n \geq 2$ ,  $a_{n+1}=(n+1)^{a_n}$ . Determine the number of integers  $k$  between  $2$  and  $2020$ , inclusive, such that  $k+1$  divides  $a_k - 1$ .

*Proposed by Taiki Aiba*

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