
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
In a bag, there are 2 white balls and $n$ red balls ($n \geq 2$, and $n \in \mathbb{N}^*$). Each time, two balls are drawn from the bag (after each draw, these two balls are put back into the bag). If the two drawn balls are of the same color, it is considered a win; otherwise, it is considered a loss.  
(1) Try to express the probability $P$ of winning in a single draw using an algebraic expression that includes $n$;  
(2) If $n=3$, calculate the probability of winning exactly once in three draws;  
(3) Let the probability of winning exactly once in three draws be $f(p)$, for what value of $n$ is $f(p)$ maximized.

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