1-2-3 type Pythagorean triangles

An integer-sided triangle is a 1-2-3 type Pythagorean triangle if its three sides a,b,c satisfy the condition : $1\times a^2+2\times b^2=3\times c^2$

For example, A triangle with three sides {$5,13,11$} is a 1-2-3 type Pythagorean triangle because $1\times 5^2+2\times {13}^2=3\times {11}^2$

Find all distinct 1-2-3 type Pythagorean triangles with perimeter $\le 25000000$

Answer format: $count,\sum perimeter$

You are given: $595,315428$ for a threshold of $1000$