What is Phi doing here?

Algebra Level 4

\[(x^2-7x+12)^{x^2-3x+2} = 1\] For positive integers \(a_1, a_2, a_3, a_4\) and \(\displaystyle \phi = \frac{1+\sqrt{5}}{2}\), if the solution of this equation is in the form of \[a_1, a_2, a_3+\phi, a_4-\phi\] Find the value of \((a_1a_2)^2+a_1a_2(a_3+1)(a_4-1)\)


This question is inspired by a series of Cedie Camomot's experiment
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