
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 certain electronic toy displays either a red or a green ball after a button is pressed. The probability of either color appearing on the first press is $\frac{1}{2}$. From the second press onwards, if a red ball appeared previously, the probabilities of a red or green ball appearing next are $\frac{1}{3}$ and $\frac{2}{3}$, respectively. If a green ball appeared previously, the probabilities of a red or green ball appearing next are $\frac{3}{5}$ and $\frac{2}{5}$, respectively. Denote the probability of a red ball appearing after the $n^{\text{th}}$ ($n \in \mathbb{N}, n \geqslant 1$) button press as $P_{n}$.

(1) Calculate the value of $P_{2}$;
(2) For $n \in \mathbb{N}, n \geqslant 2$, express $P_{n}$ in terms of $P_{n-1}$;
(3) Derive an expression for $P_{n}$ in terms of $n$.

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