
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
The sequences \(\{a_{n}\}\) and \(\{b_{n}\}\) satisfy \(a_{1}=1, b_{1}=7\), and the recursive relations \(\begin{cases} a_{n+1} = b_{n} - 2a_{n}, \\ b_{n+1} = 3b_{n} - 4a_{n} \end{cases}\). Find the value of \(\lim_{n \rightarrow \infty} \frac{a_{n}}{b_{n}}\).

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