Palindromic coin flips

It doesn't matter what you get on the first 6 flips because the palindrome starts to take shape from the $7^{th}$ flip onwards. The outcome of the $7^{th}$ flip should be same as the outcome of the $5^{th}$ flip. The probability for this is $\frac{1}{2}$ . Then the outcome of $8^{th}$ flip should be same as the outcome of the $4^{th}$ flip. Again the probability is $\frac{1}{2}$ . And so on. So the probability of getting a complete palindrome is $(\frac{1}{2})^{5} = \frac{1}{32}$ Hence, $(a+b)^2=1,089$