
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 the sequence $\{a_n\}$ that satisfies $a_1=1$, $a_2=3$, and $|a_{n+1}-a_n|=2^{n}$ $(n\in\mathbb{N}^*)$, and $\{a_{2n-1}\}$ is an increasing sequence while $\{a_{2n}\}$ is a decreasing sequence, find the limit $\lim\limits_{n\to+\infty} \dfrac{a_{2n-1}}{a_{2n}}$.

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