
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. 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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