
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
For a given integer \( a_1 (\neq -1) \), find a sequence of real numbers \(\{a_{n}\}\) \((a_i \neq 0, i=1,2,\cdots,5)\), such that if \(x_1, x_2, \cdots, x_5, y_1, y_2, \cdots, y_5\) satisfy:
$$
\begin{cases}
b_{11} x_{1} + b_{12} x_{2} + \cdots + b_{15} x_{5} = x_{1}, \\
b_{21} x_{1} + b_{22} x_{2} + \cdots + b_{25} x_{5} = x_{2}, \\
\vdots \\
b_{51} x_{1} + b_{52} x_{2} + \cdots + b_{55} x_{5} = x_{5},
\end{cases}
$$
and
$$
\begin{cases}
b_{11} y_{1} + b_{12} y_{2} + \cdots + b_{15} y_{5} = 2 y_{1}, \\
b_{21} y_{1} + b_{22} y_{2} + \cdots + b_{25} y_{5} = 2 y_{2}, \\
\vdots \\
b_{51} y_{1} + b_{52} y_{2} + \cdots + b_{55} y_{5} = 2 y_{5},
\end{cases}
$$
then \( x_1 y_1 + x_2 y_2 + \cdots + x_5 y_5 = 0 \), where \( b_{ij} = \prod_{1 \leq k \leq i} (1 + j a_k) \).

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