Time Limit: 1 s
Memory Limit: 128 MB

Submission：1
AC：1
Score：99.94

Given is a function

for a non-negative

and a non-negative integer

≤

. One can construct an infinite sequence

, where

is defined recursively as follows:

and

=

.

It is easy to see that each such sequence *F* is eventually periodic, that is periodic from some point onwards, e.g 1, 2, 7, 5, 4, 6, 5, 4, 6, 5, 4, 6 ... . Given non-negative integer *N ≤ 11000000 *, *n ≤ N* and *f*, you are to compute the period of sequence *F*.

Each line of input contains *N*, *n* and the a description of *f* in postfix notation, also known as Reverse Polish Notation (RPN). The operands are either unsigned integer constants or *N* or the variable *x*. Only binary operands are allowed: + (addition), * (multiplication) and % (modulo, i.e. remainder of integer division). Operands and operators are separated by whitespace. The operand % occurs exactly once in a function and it is the last (rightmost, or topmost if you wish) operator and its second operand is always *N* whose value is read from input. The following function:

2 x * 7 + N %

is the RPN rendition of the more familiar infix

. All input lines are shorter than 100 characters. The last line of input has

equal 0 and should not be processed.

For each line of input, output one line with one integer number, the period of *F* corresponding to the data given in the input line.

input

10 1 x N %
11 1 x x 1 + * N %
1728 1 x x 1 + * x 2 + * N %
1728 1 x x 1 + x 2 + * * N %
100003 1 x x 123 + * x 12345 + * N %
0 0 0 N %

output

1
3
6
6
369