
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  $n$  be a positive integer, and let  $S_n = \{1, 2, \ldots, n\}$ . For a permutation  $\sigma$  of  $S_n$  and an integer  $a \in S_n$ , let  $d(a)$  be the least positive integer  $d$  for which \[\underbrace{\sigma(\sigma(\ldots \sigma(a) \ldots))}_{d \text{ applications of } \sigma} = a\](or  $-1$  if no such integer exists). Compute the value of  $n$  for which there exists a permutation  $\sigma$  of  $S_n$  satisfying the equations \[\begin{aligned} d(1) + d(2) + \ldots + d(n) &= 2017,  \frac{1}{d(1)} + \frac{1}{d(2)} + \ldots + \frac{1}{d(n)} &= 2. \end{aligned}\]

*Proposed by Michael Tang*

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