PROBLEM NO.41 (B0MC-2,DAY-5)

[SANZEED]

Find the least positive integer $r$ such that for any positive integers $a,b,c,d$ ,\[((abcd)!)^{r}\] is divisible by the product :
\[\prod _{cyclic}(a!)^{bcd+1}\prod _{cyclic}((ab)!)^{cd+1}\prod _{cyclic}((abc)!)^{d+1}\].
If you can't understand,contact someone elder,I can't write all $14$ terms....sorry,really.

[SANZEED]

Hint:

First ,if you have read the chapters of $104$ nt problems,then you have read example $1.74$ ,try to use it here.

you will also need pairing tool.