Mystery Triangle

Geometry Level 5

Triangle \(X_1X_2X_3\) has \(X_1X_2=1\). Circles \(\Gamma_1\), \(\Gamma_2\), and \(\Gamma_3\) are drawn such that \(\Gamma_i\) has center \(X_i\) and passes through \(X_{i+1}\) (\(\Gamma_3\) passes through \(X_1\)).

Given that \(\Gamma_1\) is internally tangent to \(\Gamma_2\) at \(P\), and \(\Gamma_3\) also passes through \(P\), then find the value of \((X_1X_2\cdot X_2X_3\cdot X_3X_1)^2\)

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