
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
Throw a fair die (with faces numbered 1, 2, 3, 4, 5, 6) to construct a sequence $\{a_n\}$ such that $a_n= \begin{cases} 1, & \text{if an odd number appears on the } n\text{-th throw,} \\ -1, & \text{if an even number appears on the } n\text{-th throw.} \end{cases}$ Denote the sum $\sum\limits_{i=1}^{n}a_{i}$ as $a_{1}+a_{n}+\ldots+a_{n}$.
(1) Calculate the probability that $\sum\limits_{i=1}^{7}a_{i}=3$.
(2) If $\sum\limits_{i=1}^{2}a_{i}\neq 0$, calculate the probability that $\sum\limits_{i=1}^{7}a_{i}=3$.

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