Prove there exists at most one $n$-tuple $(x_1,\ldots,x_n)\in\Bbb Z^n$ satisfying the following equation:$$\dfrac{1}{x_1}+\dfrac{1}{x_2}+\cdots+\dfrac{1}{x_n}+\dfrac{1}{x_1x_2\cdots x_n}=1$$I don't know where to start. I know that the equation $x_1+\cdots+x_n=x_1x_2\cdots x_n$ has finite answers in $\Bbb N^n$ but is there any relation between these things? How can one solve such a question?
Thanks in advance!
One sequence of answers that works for every $n$ is $$ 2 $$ $$ 2, 3 $$ $$ 2,3,7 $$ $$ 2,3,7,43 $$ $$ 2,3,7,43, 1807 $$ where each new maximum entry is one added to the product of the previous entries.
If we have already solved in dimension $n-1,$ as in $S+1 = T,$ where $T$ is the product of all $n-1$ variables, appending a new $x_n = T + 1$ gives $$ x_n S + T +1 = x_n T, $$ $$ (T+1) (T-1) + T + 1 = (T+1)T, $$ $$ T^2 +T = T^2 +T $$