
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 two sequences $a_n$ and $b_n$ which satisfy $b_{n+1}a_n + b_na_{n+1} = (-1)^n+1$ and $b_n = \frac{3+(-1)^{n-1}}{2}$ for $n \in \mathbb{N}^*$, and given that $a_1 = 2$, let the sum of the first $n$ terms of the sequence $a_n$ be denoted by $S_n$. Find the value of $S_{63}$.

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