Problem 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.

趋等路径

对格点定义如下两个函数：
$$r(x,y) = (x+1,2y)$$
$$s(x,y) = (2x,y+1)$$

对于数对$(a,b)$，一条长度为$n$的趋等路径是指满足如下条件的序列$\Big((a_1,b_1),(a_2,b_2),\ldots,(a_n,b_n)\Big)$：

$(a_1,b_1) = (a,b)$
对于所有$k>1$，$(a_k,b_k) = r(a_{k-1},b_{k-1})$或$(a_k,b_k) = s(a_{k-1},b_{k-1})$
对于所有$k<n$，$a_k \ne b_k$
$a_n = b_n$

$a_n = b_n$被称为该路径的终值。

例如，
$$(45,90)\xrightarrow{r} (46,180)\xrightarrow{s}(92,181)\xrightarrow{s}(184,182)\xrightarrow{s}(368,183)\xrightarrow{s}(736,184)\xrightarrow{r}$$
$$(737,368)\xrightarrow{s}(1474,369)\xrightarrow{r}(1475,738)\xrightarrow{r}(1476,1476)$$
这是数对$(45,90)$的一条趋等路径，其长度为$10$，终值为$1476$。对于数对$(45,90)$，不存在更短的趋等路径。

对于数对$(45,90)$，求其长度为奇数的趋等路径中唯一最短的那条，并将其终值作为你的答案。