Pi and e are fine together

Calculus Level 5

Define the sequence \[x_1:=\pi+1,\quad x_{n+1}:=\pi^n+nx_n,\quad\text{for }n\geq1.\] Then we have that \[\prod_{n=1}^\infty\left(1-\frac{\pi^n}{x_{n+1}}\right)=\frac{1+\pi}{a+b\pi+e^{c\pi}}\]

for some integers \(a,b\) and \(c\). Find \[(a+b+c)^{a+b+c}.\]

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