
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
We define the sets of lattice points  $S_0,S_1,\ldots$  as  $S_0=\{(0,0)\}$  and  $S_k$  consisting of all lattice points that are exactly one unit away from exactly one point in  $S_{k-1}$ . Determine the number of points in  $S_{2017}$ .

[i]Proposed by Michael Ren

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