
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
Person A and person B each have a box. In person A's box, there are 3 red balls, 2 yellow balls, and 1 blue ball, all of the same size and shape. In person B's box, there are x red balls, y yellow balls, and z blue balls, all of the same size and shape, with x + y + z = 6 (x, y, z are positive integers). Both persons randomly pick a ball from their own box. It is determined that person B wins if they pick balls of the same color, and person A wins if they pick balls of different colors.
- $(1)$ When x = 1, y = 2, z = 3, find the probability of person B winning.
- $(2)$ If it is specified that person B scores 1 point for picking a red ball, 2 points for picking a yellow ball, and 3 points for picking a blue ball, otherwise 0 points, find the maximum average score person B can achieve, and determine the values of x, y, z at this time.

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