Marinada '21

[lang=hr] Zadan je neusmjeren graf s $2n$ čvorova, označenih redom od $1$ do $2n$, pri čemu je $n = 100$. Postoji brid između čvorova $k$ i $k + 1$ te između $n + k$ i $n + k + 1$ za sve prirodne brojeve $1 \leq k < n$. Također, postoji brid između čvorova $1$ i $n$ te između $n + 1$ i $2n$. Isto tako, bridovi povezuju $k$ i $k + n$ za sve prirodne brojeve $1 \leq k \leq n$. \\ Na koliko načina možemo obojiti svaki čvor danog grafa u jednu od $2021$ različitih boja tako da ne postoji brid čiji su vrhovi iste boje? Odgovor napišite modulo $1000000007$. [/lang] [lang=en] An undirected graph with $2n$ nodes is given, where $n = 100$. The nodes are labeled with integers from $1$ to $2n$. There is an edge between nodes $k$ and $k + 1$, and also between $n + k$ and $n + k + 1$ for every integer $1 \leq k < n$. Additionally, there is an edge between nodes $1$ and $n$, and also between $n + 1$ and $2n$. Furthermore, there is an edge between $k$ and $n + k$ for every positive integer $1 \leq k \leq n$. \\ In how many ways can we color each node in one of $2021$ different colors such that there is no edge whose nodes are the same color? Write the answer modulo $1000000007$. [/lang]