Problem 736

Paths to Equality

Define two functions on lattice points:
$r (x, y) = (x + 1, 2 y)$
$s (x, y) = (2 x, y + 1)$

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 $k > 1$
$a_{k} \neq b_{k}$ for $k < n$
$a_{n} = b_{n}$

$a_{n} = b_{n}$ is called the final value.

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

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.

趋等路径

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

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 $((a_{1}, b_{1}), (a_{2}, b_{2}), \dots, (a_{n}, b_{n}))$ ：

$(a_{1}, b_{1}) = (a, b)$
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 $k < n$ ， $a_{k} \neq b_{k}$
$a_{n} = b_{n}$

$a_{n} = b_{n}$ 被称为该路径的终值。

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