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.