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736. Paths to Equality

Define two functions on lattice points:

$r(x,y) = (x+1,2y)$ $s(x,y) = (2x,y+1)$

A path to equality of length $n$ for a pair $(a,b)$ is a sequence $\Big((a_1,b_1),(a_2,b_2),\ldots,(a_n,b_n)\Big)$, where:

• $(a_1,b_1) = (a,b)$
• $(a_k,b_k) = r(a_{k-1},b_{k-1})$ or $(a_k,b_k) = s(a_{k-1},b_{k-1})$ for $k > 1$
• $a_k \ne b_k$ for $k < n$
• $a_n = b_n$

$a_n = b_n$ is called the final value.

For example,

$(45,90)\xrightarrow{r} (46,180)\xrightarrow{s}(92,181)\xrightarrow{s}(184,182)\xrightarrow{s}(368,183)\xrightarrow{s}(736,184)\xrightarrow{r}$ $(737,368)\xrightarrow{s}(1474,369)\xrightarrow{r}(1475,738)\xrightarrow{r}(1476,1476)$

This is a path to equality for $(45,90)$ and is of length 10 with final value 1476. There is no path to equality of $(45,90)$ with smaller length.

Find the unique path to equality for $(45,90)$ with smallest odd length. Enter the final value as your answer.