
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
Toss a fair coin, with the probability of landing heads or tails being $\frac {1}{2}$ each. Construct the sequence $\{a_n\}$ such that
$$
a_n= 
\begin{cases} 
1 & \text{if the } n\text{-th toss is heads,} \\ 
-1 & \text{if the } n\text{-th toss is tails.}
\end{cases}
$$
Define $S_n =a_1+a_2+…+a_n$. 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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