
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
Sequences  $(x_n)_{n\ge1}$ ,  $(y_n)_{n\ge1}$  satisfy the relations  $x_n=4x_{n-1}+3y_{n-1}$  and  $y_n=2x_{n-1}+3y_{n-1}$  for  $n\ge1$ . If  $x_1=y_1=5$  find  $x_n$  and  $y_n$ . 

Calculate  $\lim_{n\rightarrow\infty}\frac{x_n}{y_n}$ .

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