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 $((a_1,b_1),(a_2,b_2),\dots ,(a_n,b_n))$, 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,

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

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.