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...but if p-1 hit zero we go straight up to the : on the second line of the labyrinth:

"!@ Main,Aux
" - [[n,n,0,0,0,-1,-1,a],[]]
 - top of Main is a, which is zero, so we go straight
 ! - print top of Main, which is a, which is a 0
 @ - exit the labyrinth

...but if p-1 hit zero we go straight up to the : on the second line of the labyrinth (you've seen all the instructions before, so I'm leaving their descriptions out and just giving their effect):

"!@ Main,Aux
" - [[n,n,0,0,0,-1,-1,a],[]]
 - top of Main is a, which is zero, so we go straight
 ! - print top of Main, which is a, which is a 0
 @ - exit the labyrinth

...but if the result of % was zero (for first pass only n=2 where 2%(2-1)=0) we go straight on to BOTH add the divisor to our accumulator, a=a+p, AND decrement our potential divisor, p=p-1:

;:{:{+}}""""""""{(:""} Main,Aux
; - drop top of Main [[n],[a,p]]
 : - duplicate top of Main [[n,n],[a,p]]
 { - place top of Aux onto Main [[n,n,p],[a]]
 : - duplicate top of Main [[n,n,p,p],[a]]
 { - place top of Aux onto Main [[n,n,p,p,a],[]]
 + - perform addition [[n,n,p,a+p],[]]
 } - place top of Main onto Aux [[n,n,p],[a+p]]
 } - place top of Main onto Aux [[n,n],[a+p,p]]
 """"""" - no-ops [[n,n],[a+p,p]]
 - we branch, but n is non-zero so we turn left
 " - no-op [[n,n],[a+p,p]]
 { - place top of Aux onto Main [[n,n,p],[a+p]]
 - we branch, but p is non-zero so we turn right
 ( - decrement top of Main [[n,n,p-1],[a+p]]
 : - duplicate top of Main [[n,n,p-1,p-1],[a+p]]
 "" - no-ops [[n,n,p-1,p-1],[a+p]]
 } - place top of Main onto Aux [[n,n,p-1],[a+p,p-1]]

" - no-op [[n,n,p-1],[a+p,p-1]]
 % - modulo [[n,n%(p-1)],[a+p,p-1]]
 - ...and we branch again according to the divisibility
 - of n by our new potential divisor, p-1

...but if the result of % was zero (for the first pass only when n=2) we go straight on to BOTH add the divisor to our accumulator, a=a+p, AND decrement our potential divisor, p=p-1:

;:{:{+}}""""""""{(:""} Main,Aux
; - drop top of Main [[n],[a,p]]
 : - duplicate top of Main [[n,n],[a,p]]
 { - place top of Aux onto Main [[n,n,p],[a]]
 : - duplicate top of Main [[n,n,p,p],[a]]
 { - place top of Aux onto Main [[n,n,p,p,a],[]]
 + - perform addition [[n,n,p,a+p],[]]
 } - place top of Main onto Aux [[n,n,p],[a+p]]
 } - place top of Main onto Aux [[n,n],[a+p,p]]
 """"""" - no-ops [[n,n],[a+p,p]]
 - a branch, but n is non-zero so we turn left
 " - no-op [[n,n],[a+p,p]]
 { - place top of Aux onto Main [[n,n,p],[a+p]]
 - we branch, but p is non-zero so we turn right
 ( - decrement top of Main [[n,n,p-1],[a+p]]
 : - duplicate top of Main [[n,n,p-1,p-1],[a+p]]
 "" - no-ops [[n,n,p-1,p-1],[a+p]]
 } - place top of Main onto Aux [[n,n,p-1],[a+p,p-1]]

"% Main,Aux
" - no-op [[n,n,p-1],[a+p,p-1]]
 % - modulo [[n,n%(p-1)],[a+p,p-1]]
 - ...and we branch again according to the divisibility
 - of n by our new potential divisor, p-1