grammar - Should I use its or their in the following examples?

Which sentence is right?

"All public schools provide breakfast and lunch to its students and therefore they are well nourished."

"All public schools provide breakfast and lunch to their students and therefore they are well nourished."

Answer

to their students
Issue: The number of Pronoun (its vs their)

Let's begin with the question - Student of where?
Answer: Public schools
Antecedent to which the pronoun refers to --> Schools.

Therefore, their is better than its

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Recommendations:

All public schools provide breakfast and lunch to their students. Therefore, the students are well nourished.

This school provides breakfast and lunch to its students. Therefore, the students are well nourished.