
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
Place all 15 balls numbered from 1 to 15 into three plates A, B, and C. Each plate must have at least 4 balls, and the average value of the ball numbers in plate A is 3.
$(1)$ Write down one possible set of ball numbers in plate A: ______.
$(2)$ If the average values of the ball numbers in plates B and C are 8 and 13 respectively, then the number of balls in plate B can be ______.

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