SMO 2015 Open Round 1 Question 10

Let $a_n$ be the $n$th term of a geometric progression, where $a_1=1$ and $a_3=3$. Find $$\left(\sum_{i=0}^{10}\binom{10}{i}a_{i+1}\right)\cdot\left(\sum_{j=0}^{10}(-1)^j\binom{10}{j}a_{j+1}\right).$$