Classifying a Triangle!

Geometry Level 5

In \(\triangle ABC\), suppose that the points \(M,N\) lie on the line segment \(BC\), the point \(P\) lies on the line segment \(CA\), and the point \(Q\) lies on the line segment \(AB\), such that \(MNPQ\) is a square. Suppose further that:

\[\large{\dfrac{AM}{AN} = \dfrac{AC+\sqrt{2}AB}{AB+\sqrt{2}AC}}\]

Then \(\triangle ABC\) is definitely a:

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