
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
Let \( S \) be a set of \( a + b + 3 \) points on a sphere, where \( a, b \) are nonnegative integers and no four points of \( S \) are coplanar (that is, no four points lie on a plane). Determine how many planes pass through three points of \( S \) and separate the remaining points into \( a \) points on one side of the plane and \( b \) points on the other side.

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