Whiteboard Strategy Game

In this problem, there are N participants numbered from 1 to N, each holding a whiteboard. Chairman K secretly selects one participant as the parent and the remaining as children. Chairman K writes the letter \(T\) on the parent's whiteboard and \(F\) on every child’s whiteboard. The initial assignment is hidden from the participants.

After the initial assignment, the game proceeds for L rounds according to a pre‐defined strategy. In each round (1 ≤ t ≤ L), every participant does the following simultaneously:

1. Erase the current whiteboard and write a letter, either \(T\) or \(F\). The choice can only depend on the participant’s own index, the round number, and the sequence of letters that they have read so far.
2. Choose one participant (by their number) and inform Chairman K that they want to observe that participant’s whiteboard. Chairman K then displays the chosen participant’s whiteboard so that the participant can read the letter on it.

After L rounds, every participant must guess which participant is the parent by using only the information they gathered (which includes the letter read in the initial assignment and the \(L+1\) letters observed during the game).

The objective is to design a strategy such that, regardless of who is chosen as the parent, all participants can correctly identify the parent. A smaller value of L results in a higher score.

A strategy consists of:

• A nonnegative integer L indicating the number of rounds,
• A set of rules determining for each participant what letter to write and which participant to observe at the beginning of each round (based solely on their own index, the round number, and the sequence of letters observed so far).

Additionally, after the final round, each participant must deduce the parent’s number (an integer between 1 and N) based only on the sequence of observed letters.

Your task is to design such a strategy and output, for every possible selection of the parent (i.e. parent = 1, 2, …, N), the sequence of actions taken by each participant. For each participant and for each round, output the letter written and the number of the participant they choose to observe. After all rounds, output the final guess for the parent’s number.

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Example Strategy:

One valid (though not optimal) strategy is as follows. Set L = N rounds. For every round t (1 ≤ t ≤ N):

• Each participant i writes \(T\) if and only if i is the parent and t equals i; otherwise, they write \(F\).
• Every participant observes the whiteboard of the participant with the number equal to the current round number t.

Since only the parent writes \(T\) in the round corresponding to their own index, all participants will observe exactly one \(T\) (when observing the whiteboard of participant k if the parent is k). They can then correctly deduce that the parent's number is k.

inputFormat

The input consists of a single integer N (2 ≤ N ≤ 1000) representing the number of participants.

outputFormat

For each possible parent selection (from 1 to N), output the demonstration of the actions taken by every participant. For a given parent candidate k (1 ≤ k ≤ N), the output should include:

• A header line: "Parent candidate: k".
• For each participant i (1 ≤ i ≤ N):
  • A line with "Participant i:"
  • For each round t (1 ≤ t ≤ N), a line in the format: "Round t: Write X, Observe t" where X is \(T\) if i is the parent and t = i, otherwise \(F\).
  • A final line: "Final guess: k" indicating that the participant deduces the parent's number as k.

Each parent's candidate simulation should be separated by a blank line.

sample

2

Parent candidate: 1
Participant 1:
Round 1: Write T, Observe 1
Round 2: Write F, Observe 2
Final guess: 1
Participant 2:
Round 1: Write F, Observe 1
Round 2: Write F, Observe 2
Final guess: 1
Parent candidate: 2
Participant 1:
Round 1: Write F, Observe 1
Round 2: Write F, Observe 2
Final guess: 2
Participant 2:
Round 1: Write F, Observe 1
Round 2: Write T, Observe 2
Final guess: 2

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