
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 coin, where the probabilities of getting heads or tails are both 0.5. If this coin is tossed 8 times, and for the nth toss, if heads appear, let $a_n = 1$; if tails appear, let $a_n = -1$. Define $S_n = a_1 + a_2 + \ldots + a_n$ ($1 \leq n \leq 8$). Under these conditions, calculate the following probabilities:  
(1) The probability that $S_2 \neq 0$ and $S_8 = 2$;  
(2) The probability that $S_4 = 0$ and $S_8 = 2$.

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