To the hint, Lively kind of saw, the correct solution is the word HANDS as in handsaw, since the expression hands can also be used to refer to lively or active, and so it is a nice wordplay typical of New York Times puzzles.
JIGSAW would normally be the answer to the query in the NYT Crossword, which means Lively kind of saw. The wordplay derives from the jig, a cheerful dance, and the saw, a cutting instrument. This punny, jocular use of the word is commonly applied to crosswords to give a clue to the compound word jigsaw.
