
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
Given a fair coin is tossed, the probability of heads or tails is both $\frac{1}{2}$. Construct a sequence $\{a_n\}$, where $a_n = \begin{cases} 1, \text{if heads on the nth toss} \\ -1, \text{if tails on the nth toss} \end{cases}$. Let $S_n = a_1 + a_2 + ... + a_n$. Find the probability that $S_2 \neq 0$ and $S_8 = 2$.

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