
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
Let $\xi_{0}, \xi_{1}, \ldots, \xi_{N}$ be independent Bernoulli random variables such that $\mathrm{P}\left\{\xi_{i}=1\right\}=\mathrm{P}\left\{\xi_{i}=-1\right\}=1 / 2$. Define $S_{n}=\xi_{1}+\ldots+\xi_{n}$, and let $X_{n}=\xi_{0}(-1)^{S_{n}}$ be a discrete telegraph signal for $1 \leqslant n \leqslant N$. Find the mean and variance of the random variables $X_{n}$.

Also, find the conditional distributions $\mathrm{P}\left\{X_{n}=1 \mid \xi_{0}=i\right\}, i= \pm 1$, for $1 \leqslant n \leqslant N$.

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