At Least Four Distinct Prime Factors Less Than 100

It can be verified that there are $23$ positive integers less than $1000$ that are divisible by at least four distinct primes less than $100$ .

Find how many positive integers less than $10^{16}$ are divisible by at least four distinct primes less than $100$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.