
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 defined such that $a_1=1$ and $a_na_{n+1}=\lambda \cdot 2^n$ for $n \in \mathbb{N}^*$ with $\lambda > 0$. The value of $\lambda$ is chosen so that for any $k \in \mathbb{N}^*$, the numbers $a_{2k-1}$, $a_{2k}$, and $a_{2k+1}$ form an arithmetic sequence. Find the sum of the first $20$ terms, $S_{20}$, of the sequence $\{a_n\}$.

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