
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
Xiao Zhang has a box containing $a$ red balls, $b$ yellow balls, and $c$ white balls, with $a+b+c=6$ ($a$, $b$, $c$ are natural numbers). Xiao Liu has a box containing 3 red balls, 2 yellow balls, and 1 white ball. Both of them draw one ball from their own box. It is stipulated that Xiao Zhang wins if both balls are of the same color, and Xiao Liu wins if the balls are of different colors.  
(1) Express the probability of Xiao Zhang winning in terms of $a$, $b$, and $c$.  
(2) If it is further stipulated that Xiao Zhang scores 1 point for winning with a red ball, 2 points for a yellow ball, and 3 points for a white ball, and 0 points otherwise, find the maximum expected value of Xiao Zhang's score and the corresponding values of $a$, $b$, and $c$.

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