Let \( n \) be a positive integer. Suppose that \( a_1, a_2, \dots, a_n \) are non-zero integers and \( b_1, b_2, \dots, b_n\) are positive integers such that \( (b_i, b_n) = 1 \) for \( i = 1, 2, \dots, n-1 \). Prove that the Diophantine equation
\[
a_1 x_1^{b_1} + a_2 x_2^{b_2} + \dots + a_n x_n^{b_n} = 0
\]
has infinitely many integer solutions \( (x_1, x_2, \dots, x_n) \).