--description--

Let $n$ be a positive integer. Suppose there are stations at the coordinates $(x, y) = (2^i\bmod n, 3^i\bmod n)$ for $0 ≤ i ≤ 2n$. We will consider stations with the same coordinates as the same station.

We wish to form a path from (0, 0) to ($n$, $n$) such that the $x$ and $y$ coordinates never decrease.

Let $S(n)$ be the maximum number of stations such a path can pass through.

For example, if $n = 22$, there are 11 distinct stations, and a valid path can pass through at most 5 stations. Therefore, $S(22) = 5$. The case is illustrated below, with an example of an optimal path:

It can also be verified that $S(123) = 14$ and $S(10\,000) = 48$.

Find $\sum S(k^5)$ for $1 ≤ k ≤ 30$.

--hints--

uphillPaths() should return 9936352.

assert.strictEqual(uphillPaths(), 9936352);

--seed--

--seed-contents--

function uphillPaths() {

  return true;
}

uphillPaths();

--solutions--

// solution required