
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
Eight numbers \( a_{1}, a_{2}, a_{3}, a_{4} \) and \( b_{1}, b_{2}, b_{3}, b_{4} \) satisfy the relations

$$
\left\{
\begin{array}{c}
a_{1} b_{1} + a_{2} b_{3} = 1 \\
a_{1} b_{2} + a_{2} b_{4} = 0 \\
a_{3} b_{1} + a_{4} b_{3} = 0 \\
a_{3} b_{2} + a_{4} b_{4} = 1
\end{array}
\right.
$$

It is known that \( a_{2} b_{3} = 7 \). Find \( a_{4} b_{4} \).

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