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Optimal Skiing Route

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Optimal Skiing Route
Dynamic Programming

Imagine that a skiier (S) is skiing down a slope depicted as an N x N matrix of characters. The skiier starts at the top of the slope (top-left square) and can only go downhill (down or to the right). In the skiier’s path, there is regular snow (.), obstacles (O), and flags (F). Passing through regular snow or flags takes 1 second on each square, while obstacles require slowing down to avoid and take 3 seconds to pass through. Each flag is collected when passed through. Upon reaching the bottom of the slope (bottom-right square) the player receives 5 points for every flag collected. The player also loses 1 point for every second slower they take to traverse the course than a specified pace time, though if they have a time faster than pace time, they gain one point for every second they travel faster than pace time.

Return the highest score possible, and the number of flags collected on that run. If there are multiple runs with the same highest score, pick one with the maximum number of flags. A dynamic programming solution should have runtime O(n^2).

Input format: [integer N: the size of the matrix] [integer pace: pace time in seconds] [N X N matrix of characters (., S, O, or F), without spaces, each row on a separate line]

Output format: [integer score: the total score] [integer flags: the number of flags collected]

Examples

Example input
4
6
S…
.OF.
.O.O
….
Example output:
5 1

Explanation: In this case, the pace time 6 is the time taken by the fastest route: the skiier grabs the flag while avoiding obstacles: each step depicted by a number below:

S12
.O3.
.O4O
..56

Thus the player gets 5 points for one flag, and loses no points for being slower than pace.

Example input
3
4
SOF
..F

Example output:
8 2

Explanation: In this case, the optimal route is not the fastest route, since the 10 points earned by flags offsets the 2 additional seconds caused by the obstacle.
Steps depicted below:

S12
..3
..4

The final score is 8, for 10 points from the two flags, and -2 points for requiring 6 seconds to traverse (2 more than the pace time).

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