
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 $\{a_n\}$ with its sum of the first $n$ terms denoted as $S_n$. The sequence satisfies the relationship $a_{n+2} - a_{n} = d$ ($d$ is a non-zero constant, $n \in \mathbb{N}^*$), $a_1 = 1$, $a_2 = 2$, and the sequences $a_1a_2$, $a_2a_3$, $a_3a_4$ form an arithmetic progression. Find the value of $S_{20}$.

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