Integer Right Triangles

If $p$ is the perimeter of a right angle triangle with integral length sides, ${a, b, c}$ , there are exactly three solutions for $p = 120$ .

${20, 48, 52}$ , ${24, 45, 51}$ , ${30, 40, 50}$

For which value of $p \leq 1000$ , is the number of solutions maximised?

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.