
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system

Setting up a table for a tournament in which all $n$ teams compete against each other over $n-1$ rounds is a complex mathematical problem with various solution methods. In this question, we will explore one of these approaches.

Consider a tournament with 6 teams. We associate the numbers 1, 2, 3, 4, 5, and $\infty$ (infinity) with each of the teams. The first round of the tournament is $1 \times \infty, 2 \times 5, 3 \times 4$. To set up round $i$, we add $i-1$ to each number involved in the matches of the initial round, considering the following rules:

- When the sum exceeds 5, subtract 5 from the result;
- $\infty$ added to any positive integer remains $\infty$. For example, the second round will be:

$$
\begin{gathered}
(1+1) \times(\infty+1), \text {i.e., } 2 \times \infty, \\
(2+1) \times(5+1), \text {i.e., } 3 \times 1, \\
(3+1) \times(4+1), \text {i.e., } 4 \times 5.
\end{gathered}
$$

(a) Determine the 3 remaining rounds of the tournament, following the method described above.

(b) Using this method, display the 7 rounds of a tournament with 8 teams.

Present the answer in LaTex format: \boxed{Your answer}
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