
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 $\{a_n\}$ satisfying 
$$
a_n = 
\begin{cases} 
2a_{n-1}-2, & n=2k+1 \\
a_{n-1}+1, & n=2k 
\end{cases}
(k\in\mathbb{N}^*)
$$
and $a_1 = 1$, find the sum of the first 20 terms, $S_{20}$.

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