
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 a positive integer $n$, a sequence $(a\_1, a\_2, \ldots, a_{2n})$ satisfies the following conditions:
1. $a\_i \in \{1, -1\}$, for $i = 1, 2, \ldots, 2n$;
2. For any $1 \leq k \leq l \leq n$, we have $\left|\sum\_{i=2k-1}^{2l} a\_i\right| \leq 2$.

(1) Let $A\_n$ denote the number of sequences $(a\_1, a\_2, \ldots, a_{2n})$ that satisfy "for any $1 \leq k \leq n$, we have $a_{2k-1} + a_{2k} = 0$". Find $A\_n$;

(2) Let $B\_n$ denote the number of sequences $(a\_1, a\_2, \ldots, a_{2n})$ that satisfy "there exists $1 \leq k \leq n$ such that $a_{2k-1} + a_{2k} \neq 0$". Find $B\_n$.

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