
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 bag contains $n$ red balls ($n\geqslant 5$ and $n\in N$) and $5$ white balls. Two balls are drawn from the bag at a time, and if the colors of the two balls are different, it is considered a win.
(I) Express the probability $p$ of winning in one draw using $n$;
(II) If $n=5$, find the probability of winning exactly once in three draws (with replacement);
(III) Let $P$ denote the probability of winning exactly once in three draws (with replacement). For what value of $n$ is $P$ at its maximum?

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