
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]


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Let \(\xi_1, \ldots, \xi_n\) be independent identically distributed random variables, with \(\mathrm{P}\{\xi_k=1\} = \mathrm{P}\{\xi_k=-1\}=1/2\) (a symmetric Bernoulli scheme). Define \(S_n=\xi_1 + \ldots + \xi_n\) and \(P_n(k) = \mathrm{P}\{S_n = k\}\), where \(k \in E_n = \{0, \pm 1, \ldots, \pm n\}\).

Show, based on the total probability formula, that the following recurrence relations hold (a special case of the Kolmogorov-Chapman equations, B1.I.12):

\[ P_{n+1}(k) = \frac{1}{2} P_n(k+1) + \frac{1}{2} P_n(k-1), \quad k \in E_{n+1}, \]

which are equivalent to:

\[ P_{n+1}(k) - P_n(k) = \frac{1}{2}\left[P_n(k+1) - 2 P_n(k) + P_n(k-1)\right]. \]

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