
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
Let \( n \) be an integer greater than 3. Let \( R \) be the set of lattice points \( (x, y) \) such that \( 0 \leq x, y \leq n \) and \( |x-y| \leq 3 \). Let \( A_n \) be the number of paths from \( (0,0) \) to \( (n,n) \) that consist only of steps of the form \( (x, y) \rightarrow (x, y+1) \) and \( (x, y) \rightarrow (x+1, y) \) and are contained entirely within \( R \). Find the smallest positive real number that is greater than \( \frac{A_{n+1}}{A_n} \) for all \( n \).

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