
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 company received an order to produce 3000 units of a product, which consists of three parts: A, B, and C. Each unit requires 2, 2, and 1 pieces of parts A, B, and C, respectively. It is known that each worker can produce 6 pieces of part A, 3 pieces of part B, or 2 pieces of part C per day. The company plans to divide 200 workers into three groups to produce these three parts. The number of workers producing part B is proportional to the number of workers producing part A, with a proportionality constant of $K$ ($K$ is a positive integer).
(1) Let the number of workers producing part A be $x$. Write down the time required to complete the production of parts A, B, and C, respectively.
(2) Assuming the production of these three parts starts at the same time, determine the value of the positive integer $K$ to minimize the completion time of the order, and provide the specific grouping of workers when the completion time is shortest.

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