Eventually Periodic Sequence

For a non-negative integer $N$, a function $f\colon \{0, 1, \dots, N\} \to \{0, 1, \dots, N\}$ is given, together with a non-negative integer $n \le N$. From these one can build the infinite sequence $F = f^{1}(n), f^{2}(n), \dots, f^{k}(n), \dots$, where $f^{k}(n)$ is defined recursively by $f^{1}(n) = f(n)$ and $f^{k+1}(n) = f(f^{k}(n))$.

Every such sequence $F$ is eventually periodic: from some point onward it repeats, for example $1, 2, 7, 5, 4, 6, 5, 4, 6, 5, 4, 6, \dots$.

Given $N \le 11000000$, $n \le N$, and the function $f$, compute the period of the sequence $F$.

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

2 x * 7 + N %

is the more familiar infix $(2 \* x + 7) \% N$. Every input line is shorter than 100 characters. The last line of input has $N = 0$ and must not be processed.

For each line of input, output one line containing a single integer: the period of the sequence $F$ that corresponds to the data on that input line.