
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
A fun "Sports Day" organized by a certain school for seventh graders has a total of five competition events. Each class participates in all events, and it is stipulated that only the top three classes in each event will be scored (no tied classes). The class that comes in first place is awarded $a$ points, the second place class is awarded $b$ points, and the third place class is awarded $c$ points ($a \gt b \gt c$, $a$, $b$, and $c$ are all positive integers). The total score for each class in the competition is the sum of the scores in each event. There are four classes in this grade. If the total scores for these four classes in the "Sports Day" are $21$, $6$, $9$, and $4$ respectively, then $a+b+c=$______, and the value of $a$ is ______.

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