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문제

Your task in this problem is to find out the minimum number of stones needed to place on an N-by-M rectangular grid (N horizontal line segments and M vertical line segments) to enclose at least K intersection points. An intersection point is enclosed if either of the following conditions is true:

  1. A stone is placed at the point.
  2. Starting from the point, we cannot trace a path along grid lines to reach an empty point on the grid border through empty intersection points only.

For example, to enclose 8 points on a 4x5 grid, we need at least 6 stones. One of many valid stone layouts is shown below. Enclosed points are marked with an "x".

입력

The first line of the input gives the number of test cases, T. T lines follow. Each test case is a line of three integers: N M K.

Limits

  • 1 ≤ T ≤ 100.
  • 1 ≤ N.
  • 1 ≤ M.
  • 1 ≤ KN × M.
  • N × M ≤ 1000.

출력

For each test case, output one line containing "Case #x: y", where x is the test case number (starting from 1) and y is the minimum number of stones needed.

예제 입력 1

2
4 5 8
3 5 11

예제 출력 1

Case #1: 6
Case #2: 8

힌트