
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
Eighteen students participate in a team selection test with three problems, each worth up to seven points. All scores are nonnegative integers. After the competition, the results are posted by Evan in a table with 3 columns: the student's name, score, and rank (allowing ties), respectively. Here, a student's rank is one greater than the number of students with strictly higher scores (for example, if seven students score  $0, 0, 7, 8, 8, 14, 21$  then their ranks would be  $6, 6, 5, 3, 3, 2, 1$  respectively).

When Richard comes by to read the results, he accidentally reads the rank column as the score column and vice versa. Coincidentally, the results still made sense! If the scores of the students were  $x_1 \le x_2 \le \dots \le x_{18}$ , determine the number of possible values of the  $18$ -tuple  $(x_1, x_2, \dots, x_{18})$ . In other words, determine the number of possible multisets (sets with repetition) of scores.

*Proposed by Yang Liu*

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