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Problem Statement
There are n vertices forming a simple cycle. It is guaranteed that the graph satisfies the following property:
- There exists a permutation p of length n such that for every i (1 ≤ i < n), the vertices p_i and p_{i+1} are adjacent in the graph, and also p_n and p_1 are adjacent.
At the beginning of the interaction, a token is placed at a predetermined starting vertex s (1 ≤ s ≤ n).
The value of n is hidden from the contestant. The contestant may interact with the judge in order to determine n.
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Interaction Protocol
The contestant may issue the following commands:
1. walk x
- Input: a nonnegative integer x (0 ≤ x ≤ 10^9).
- Effect: the token moves x steps forward along the cycle from its current position.
- Output: the label of the vertex reached after the move.
2. guess g
- Input: an integer g (1 ≤ g ≤ 10^9).
- Effect: ends the interaction.
- If g = n, the answer is considered correct. Otherwise, it is considered wrong.
Constraints:
- 1 ≤ n ≤ 10^9
- 1 ≤ s ≤ n
- The number of walk operations must not exceed 200000.
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Scoring
Let q be the number of walk operations made before the guess.
- If the guess is wrong or q > 200000, the score is 0.
- Otherwise, the score is f(q), where f is a continuous, monotonically decreasing function defined in log10-space by linear interpolation through the following anchor points:
f(1) = 100
f(10000) = 95
f(20000) = 60
f(50000) = 30
and further decreasing linearly to
f(200000) = 0.
Thus, fewer walk queries yield a higher score, with a perfect solution scoring close to 100.
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