
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 sequence $\{a_n\}$ is an infinite sequence. If $\{a_n\}$ represents the sum of the coefficients of each term in the binomial expansion of $(1+2x)^n$ ($n\in\mathbb{N}^+$), then $\lim_{n\to\infty} \left(\frac{1}{a_1} + \frac{1}{a_2} + \ldots + \frac{1}{a_n}\right) = \_\_\_\_\_$.

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