Binary Strings are Fun

A binary string$$$^\dagger$$$ $$$b$$$ of odd length $$$m$$$ is good if $$$b_i$$$ is the median$$$^\ddagger$$$ of $$$b[1,i]^\S$$$ for all odd indices $$$i$$$ ($$$1 \leq i \leq m$$$).

For a binary string $$$a$$$ of length $$$k$$$, a binary string $$$b$$$ of length $$$2k-1$$$ is an extension of $$$a$$$ if $$$b_{2i-1}=a_i$$$ for all $$$i$$$ such that $$$1 \leq i \leq k$$$. For example, 1001011 and 1101001 are extensions of the string 1001. String $$$x=$$$1011011 is not an extension of string $$$y=$$$1001 because $$$x_3 \neq y_2$$$. Note that there are $$$2^{k-1}$$$ different extensions of $$$a$$$.

You are given a binary string $$$s$$$ of length $$$n$$$. Find the sum of the number of good extensions over all prefixes of $$$s$$$. In other words, find $$$\sum_{i=1}^{n} f(s[1,i])$$$, where $$$f(x)$$$ gives number of good extensions of string $$$x$$$. Since the answer can be quite large, you only need to find it modulo $$$998\,244\,353$$$.

$$$^\dagger$$$ A binary string is a string whose elements are either $$$\mathtt{0}$$$ or $$$\mathtt{1}$$$.

$$$^\ddagger$$$ For a binary string $$$a$$$ of length $$$2m-1$$$, the median of $$$a$$$ is the (unique) element that occurs at least $$$m$$$ times in $$$a$$$.

$$$^\S$$$ $$$a[l,r]$$$ denotes the string of length $$$r-l+1$$$ which is formed by the concatenation of $$$a_l,a_{l+1},\ldots,a_r$$$ in that order.