Fantastic Ratios

Level pending

Let \(ABCD\) be a square and \(N\) a point on segment \(CD\) such that \(AD = 3 ND\). Let \(M\) be a point on segment \(BC\) satisfying \(\angle MAN = 45^\circ\). The ratio \(\frac{AM}{AN}\) can be expressed as \(\frac{a\sqrt{b}}{c}\), where \(a, b,\) and \(c\) are positive integers, \(a\) and \(b\) are relatively prime, and \(c\) is squarefree. Find the value of \(ac - b^2\).

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