
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 Cartesian plane, let \( A=(0,0) \), \( B=(200,100) \), and \( C=(30,330) \). Compute the number of ordered pairs \( (x, y) \) of integers such that \( \left(x+\frac{1}{2}, y+\frac{1}{2}\right) \) is in the interior of triangle \( ABC \).

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