A12092 | Sophisticated Device

You are given integers $d$ and $p$ , $p$ is prime.

Also you have a mysterious device. It has memory cells, each contains an integer between $0$ and $p-1$ . Also two instructions are supported, addition and raising to the $d$ -th power. $\textbf{Both are modulo}$ $p$ .

The memory cells are numbered $1, 2, \dots, 5000$ . Initially cells $1$ and $2$ contain integers $x$ and $y$ , respectively ( $0 \leqslant x, y \leq p - 1$ ). All other cells contain $\textbf{1}$ s.

You can not directly access values in cells, and you $\textbf{don't know}$ values of $x$ and $y$ (but you know they are written in first two cells). You mission, should you choose to accept it, is to write a program using the available instructions to obtain the product $xy$ modulo $p$ in one of the cells. You program should work for all possible $x$ and $y$ .

Addition instruction evaluates sum of values in two cells and writes it to third cell. This instruction is encoded by a string "+ e1 e2 to", which writes sum of values in cells e1 and e2 into cell to. Any values of e1, e2, to can coincide.

Second instruction writes the $d$ -th power of a value in some cell to the target cell. This instruction is encoded by a string "^ e to". Values e and to can coincide, in this case value in the cell will be overwritten.

Last instruction is special, this is the return instruction, and it is encoded by a string "f target". This means you obtained values $xy \bmod p$ in the cell target. No instructions should be called after this instruction.

Provide a program that obtains $xy \bmod p$ and uses no more than $5000$ instructions (including the return instruction).

It is guaranteed that, under given constrains, a solution exists.

The first line contains space-separated integers $d$ and $p$ ( $2 \leqslant d \leqslant 10$ , $d < p$ , $3 \leqslant p \leqslant 10^9 + 9$ , $p$ is prime).

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