Path Sum: Four Ways

NOTE: This problem is a significantly more challenging version of Problem 81.

In the \( 5 \) by \( 5 \) matrix below, the minimal path sum from the top left to the bottom right, by moving left, right, up, and down, is indicated in bold red and is equal to \( 2297 \).

\[ \begin{pmatrix} \color{red}{131} & 673 & \color{red}{234} & \color{red}{103} & \color{red}{18}\\ \color{red}{201} & \color{red}{96} & \color{red}{342} & 965 & \color{red}{150}\\ 630 & 803 & 746 & \color{red}{422} & \color{red}{111}\\ 537 & 699 & 497 & \color{red}{121} & 956\\ 805 & 732 & 524 & \color{red}{37} & \color{red}{331}\\ \end{pmatrix} \]

Find the minimal path sum from the top left to the bottom right by moving left, right, up, and down in matrix.txt (right click and "Save Link\( / \)Target As\( ... \)"), a \( 31 \)K text file containing an gc\( 80 \) by \( 80 \) matrix.