
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
Let \( x \) be the number of 7th graders who together scored \( n \) points in the tournament. The number of 8th graders participating in the tournament is \( 10 \times x \), and they scored a total of \( 4.5 \times n \) points. Thus, the total number of students participating in the tournament is \( 11 \times x \), and they scored \( 5.5 \times n \) points collectively. The total number of points scored by all participants is equal to the number of games played. We obtain the equation:

$$
\begin{gathered}
5.5 \times n = \frac{11 \times x \times (11x - 1)}{2} \\
n = x \times (11x - 1)
\end{gathered}
$$

Each 7th grader played \( 11x - 1 \) games (since there are \( 11x \) participants in total). Therefore, \( x \) 7th graders could collectively score \( n \) points, i.e., \( x \times (11x - 1) \) points, only if each of them won all their games. This is possible only when \( x = 1 \) (since two 7th graders cannot simultaneously win against each other).

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