
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
If the sum of the coefficients in the binomial expansion of $(1+2x)^{n}$ (where $n \in \mathbb{N}^{*}$) is $a_n$, and the sum of the binomial coefficients is $b_n$, then determine the limit:
$$ \lim_{n \to \infty} \frac{b_{n+1} - a_{n}}{a_{n+1} + b_{n}} = \text{_____} .$$

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