Solving infinite system of polynomial equations with equal powers

Question

I have the following infinite system with unknown $\{\mu_i\}_{i=1}^k$, $$ \mu_1^{\ell} a_1(\ell) + \mu_2^{\ell} a_2(\ell) + \ldots + \mu_k^{\ell} a_k(\ell) = b(\ell), $$ for $\ell \in \mathbb{N}_{>0}$, $a_i(\ell) \neq 0$ and $b(\ell) \neq 0$.

1. Under which conditions is this system solvable?
2. When this system is solvable can the $\{\mu_i\}_{i=1}^k$ be given explicitly?

Best regards,