
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 \( n \in \mathbf{Z}_{+}, \) let \( A=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\} \) be a sequence consisting of numbers 0 or 1, where \( a_{k}=0 \) or \( 1 \) for \( 1 \leq k \leq n \).

(1) If \( n \geq 3 \), define another sequence \( A^{\prime}=\left\{a_{1}^{\prime}, a_{2}^{\prime}, \cdots, a_{n}^{\prime}\right\} \) from the sequence \( A \) where:
\[
a_{k}^{\prime}=\left\{\begin{array}{ll}
0, & \text{if } a_{k-1}=a_{k+1} ; \\
1, & \text{if } a_{k-1} \neq a_{k+1},
\end{array}\right.
\]
and given:
\[ 
k=1,2, \cdots, n, \quad a_{0}=a_{n}, \quad a_{n+1}=a_{1}.
\]

Find all sequences \( A \) such that \( a_{k} + a_{k}^{\prime} = 1 \) for \( k=1,2, \cdots, n \) (write only the results, no need to show the solving process).

(2) Determine the number of sequences \( A \) such that the sum \( a_{1} + a_{2} + \cdots + a_{n} \) leaves a remainder of 3 when divided by 4.

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