Is slept a action verb

No, the Word 'slept' is the past tense of the verb to sleep.

• My brother slept on the couch when he visited.

The Word 'sleep' is both a verb and a noun.

The noun 'sleep' is a common noun; a general Word for the natural periodic state of rest for the mind and body; a period of this form of rest; a state of a loss of feeling in a part of the body, followed by tingling; a Word for any sleep or period of sleep.