does jogging / HIIT require extra protein?

I know that with weight lifting you are supposed to increase your protein intake to help muscle growth and repair, but should you do the same for jogging / HIIT?

should protein be eaten with breakfast right after a HIIT session?

also, what happens to protein if it's not used up in your body? Will it get stored and increase your weight / fatness if you eat too much?
 
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