## Paths to Equality

### Problem 736

Define two functions on lattice points:

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,

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.