We can approach this problem by realizing that we can reach any integer n by making repeated jumps of size 1, alternating between moving to the left and moving to the right. Specifically, to reach n, we can make the following jumps:

1. Move 1 unit to the right
2. Move 2 units to the left
3. Move 3 units to the right
4. Move 4 units to the left
5. Move 5 units to the right
6. Move 6 units to the left
7. and so on, continuing to alternate directions and increasing the size of the jump by 1 each time

This sequence of jumps will eventually reach n, since the sum of the jump sizes is 1 – 2 + 3 – 4 + 5 – 6 + … = 1 + (3 – 2) + (5 – 4) + (7 – 6) + … is an infinite series that diverges to infinity.

To find the optimal solution to reach a given number n, we can first check if it is reachable by computing the sum of the first k integers, where k is the smallest positive integer such that n is less than or equal to the sum. If n is not reachable, we can stop and return “not reachable”. Otherwise, we can find the sequence of jumps as follows:

1. Start at 0
2. Move 1 unit to the right
3. Move 2 units to the left, unless we have already reached n
4. Move 3 units to the right, unless we have already reached n
5. Move 4 units to the left, unless we have already reached n
6. Continue alternating between left and right, increasing the jump size by 1 each time, until we reach n.

This algorithm will terminate after at most 2n steps, since the sum of the jump sizes is at most 1 + 2 + … + n = n(n+1)/2. Therefore, it provides an optimal solution to reach a given number if it is reachable.