Is the fast valid for someone who forgot to make the intention overnight and woke up after 5:00 AM, then intended to fast and completed his day without eating or drinking? And is it permissible to swallow phlegm during fasting if the fasting person fears that his fast will be invalidated by expelling it?

"Formulating the intention (niyyah) at night is a pillar of fasting according to the majority of scholars, based on the Prophet's (peace be upon him) saying: 'Whoever does not formulate the intention to fast from the night, there is no fast for him.' It is sufficient to merely intend to fast at night, such as by taking the pre-dawn meal (suhoor) or by eating and drinking with the intention of gaining strength for the fast. However, if the intention is not formulated at night, then making up that day is obligatory. Swallowing phlegm after it has reached the mouth invalidates the fast according to many scholars. However, if it does not reach the outer part of the mouth, then swallowing it does not invalidate the fast."