(Generalized inverse method) Let $\{a_n\}$ and $\{b_n\}$ be two given sequence and $p$ be a non-negative integer. Show that the following two relationships are equivalent $$a_n=\sum_{k=0}^{n}(-1)^k\binom{n+p}{k+p}b_k\Leftrightarrow b_n=\sum_{k=0}^{n}(-1)^k\binom{n+p}{k+p}a_k$$