The question refers to "the reaction given below", but the actual reaction scheme/image is missing from the prompt.

Without the structure of the reactant(s), reagent(s), and product pathway, the major product $P$ cannot be determined, so the number of atoms with $sp^2$ hybridization in $P$ also cannot be computed uniquely.

What is needed

Please provide the reaction diagram or write the reaction explicitly, including:

1. Reactant structure
2. Reagents/conditions
3. Any intermediate/product labels

Why the stored answer cannot be verified yet

The stored correct answer is given as $8$ or $12$, which suggests the missing figure may have had multiple acceptable interpretations or versions. But without the reaction, I cannot logically derive either value.

Conclusion

At present, the problem is incomplete, so no valid step-by-step chemical solution can be produced.