Implement a function that locates key in an increasing sequence you can only read one element at a time through the callback at, spending at most 24 reads.

• There is no array. at(i) hands you the value stored at index i, and it is the only way to see the sequence. Grademe counts every call, so each one costs you something.
• The sequence holds n values and is already sorted in non-decreasing order: at(0) <= at(1) <= ... <= at(n - 1). Take that as given, never check it.
• Call at only with an index inside [0, n). An index of n or beyond is an out of range read even when it looks harmless.
• Return the index of key as a long. The same value may appear several times in a row, and the answer is the smallest index holding it, never a later one.
• An n of 0 means there is nothing to look at. Return before calling at a single time.
• Budget: 24 calls to at per search, for any n up to 1000000. Walking the indices from 0 upward answers every short sequence and then burns half a million reads on a long one. Discard half of what is left on each read instead.
• Grademe prints the value you returned next to what the search cost: idx=3 probes<=24 when the call stayed inside the budget. Go over it and the real count is printed instead, as in probes=1000000. Read outside [0, n) and it prints probes=out-of-range.
• Return -1 when no index holds key. A key below at(0), a key above at(n - 1), and a key that falls between two neighbours all end the same way.

Implement a function that locates key in an increasing sequence you can only read one element at a time through the callback at, spending at most 24 reads.

• There is no array. at(i) hands you the value stored at index i, and it is the only way to see the sequence. Grademe counts every call, so each one costs you something.
• The sequence holds n values and is already sorted in non-decreasing order: at(0) <= at(1) <= ... <= at(n - 1). Take that as given, never check it.
• Call at only with an index inside [0, n). An index of n or beyond is an out of range read even when it looks harmless.
• Return the index of key as a long. The same value may appear several times in a row, and the answer is the smallest index holding it, never a later one.
• An n of 0 means there is nothing to look at. Return before calling at a single time.
• Budget: 24 calls to at per search, for any n up to 1000000. Walking the indices from 0 upward answers every short sequence and then burns half a million reads on a long one. Discard half of what is left on each read instead.
• Grademe prints the value you returned next to what the search cost: idx=3 probes<=24 when the call stayed inside the budget. Go over it and the real count is printed instead, as in probes=1000000. Read outside [0, n) and it prints probes=out-of-range.
• Return -1 when no index holds key. A key below at(0), a key above at(n - 1), and a key that falls between two neighbours all end the same way.