
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
A person flips a coin, where the probability of heads up and tails up is $\frac{1}{2}$ each. Construct a sequence $\left\{a_{n}\right\}$ such that
$$
a_{n}=\left\{
\begin{array}{ll}
1, & \text{if the } n \text{th flip is heads;} \\
-1, & \text{if the } n \text{th flip is tails.}
\end{array}
\right.
$$
Let $S_{n}=a_{1}+a_{2}+\cdots+a_{n}$. Find the probability that $S_{2} \neq 0$ and $S_{8}=2$. Provide your answer in its simplest fractional form.

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