Almost Right-angled Triangles II

Let us call an integer sided triangle with sides $a \leq b \leq c$ barely obtuse if the sides satisfy
$a^{2} + b^{2} = c^{2} - 1$ .

How many barely obtuse triangles are there with perimeter $\leq 75 000 000$ ?

Write-up coming later

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