
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

Some students from the seventh and eighth grades of a school participate in a ping-pong tournament, where each student plays against all other students exactly once, earning 1 point for a win and 0 points for a loss. There are ten times more eighth-grade students than seventh-grade students. The total score of the eighth-grade students is 4.5 times the total score of the seventh-grade students.

a) Verify that if there are \( k \) students in the tournament, then the number of games is \( \frac{k(k-1)}{2} \).

b) What is the sum of the scores obtained by all the seventh-grade students?

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