
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 $\xi_{1}, \ldots, \xi_{N}$ be independent Bernoulli random variables with $\mathrm{P}\left\{\xi_{i}=1\right\}=p$ and $\mathrm{P}\left\{\xi_{i}=-1\right\}=1-p$. Define \( S_{i}=\xi_{1}+\ldots+\xi_{i} \) for \( 1 \leqslant i \leqslant N \) and \( S_{0}=0 \). Let \(\mathscr{R}_{N}\) be the number of distinct points visited by the random walk \( S_{0}, S_{1}, \ldots, S_{N} \).

Find \(\mathrm{E} \mathscr{R}_{N}\). Determine the values of \( p \) for which the law of large numbers holds for \(\mathscr{R}_{N}\) in the form

$$
\mathrm{P}\left\{\left|\frac{\mathscr{R}_{N}}{N}-c\right|>\varepsilon\right\} \rightarrow 0, \quad N \rightarrow \infty
$$

where \(\varepsilon>0\) and \(c\) is some constant.

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