
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
How many lattice points are exactly twice as close to  $(0,0)$  as they are to  $(15,0)$ ? (A lattice point is a point  $(a,b)$  such that both  $a$  and  $b$  are integers.)

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