
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
Given a sequence $\left\{a_n\right\}$ satisfying $a_1=1, a_2=3$, if $\left|a_{n+1}-a_n\right|=2^n$ $(n\in\mathbb{N}^*)$, and $\left\{a_{2n-1}\right\}$ is an increasing sequence, $\left\{a_{2n}\right\}$ is a decreasing sequence, then find $\lim\limits_{n\to\infty} \frac{a_{2n-1}}{a_{2n}}$.

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