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#JavaScript (ES6), 25 bytes

Takes input as (k)(n).

k=>g=n=>n>k?n*g(n-k):n|!n 

Try it online!

Or 26 bytes to support BigInts.

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#JavaScript (ES6), 59 bytes

Takes input as a string, using the format described in the challenge.

s=>(a=s.split`!`,k=a.length,g=n=>n*=n<k||g(n-k+1))(a[0])||1 

Try it online!