Let $N$ be the sum of all binomial coefficients $\binom{a}{b}$ such that $a$ and $b$ are nonnegative integers and $a+b$ is an even integer less than 100.  Find the remainder when $N$ is divided by 144.  (Note: $\binom{a}{b} = 0$ if $a<b$, and $\binom{0}{0} = 1$.)

Let's think step by step and output the final answer within \boxed{}.
user