
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
In the coordinate plane, points with integer values for both coordinates are called lattice points. For a certain lattice point \( P \) and a positive number \( d \), if there are exactly \( k(>0) \) distinct lattice points at a distance \( d \) from \( P \), the range of values for \( k \) is denoted as \( \left\{k_1, k_2, \cdots\right\} \) where \( 0<k_1<k_2<\cdots \). What is \( k_2 \)?

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