
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
Suppose that  $x_1 < x_2 < \dots < x_n$  is a sequence of positive integers such that  $x_k$  divides  $x_{k+2}$  for each  $k = 1, 2, \dots, n-2$ .  Given that  $x_n = 1000$ , what is the largest possible value of  $n$ ?

*Proposed by Evan Chen*

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