Let ${\mathbb{R}}^2$ be the set of pairs of real numbers $(x,y)$. Let $\pi =3.14159\cdots$.

Consider the function $f$ from ${\mathbb{R}}^2$ to ${\mathbb{R}}^2$ defined by $f(x,y)=(x^2-x-y^2,2xy-y+\pi )$, and its $n$-th iterated composition $f^{(n)}(x,y)=f(f(\cdots f(x,y)\cdots ))$. For example $f^{(3)}(x,y)=f(f(f(x,y)))$. A pair $(x,y)$ is said to have period $n$ if $n$ is the smallest positive integer such that $f^{(n)}(x,y)=(x,y)$.

Let $P(n)$ denote the sum of $x$-coordinates of all points having period not exceeding $n$. Interestingly, $P(n)$ is always an integer. For example, $P(1)=2$, $P(2)=2$, $P(3)=4$.

Find $P({10}^7)$ and give your answer modulo $1020340567$.