
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 $a_n \neq 0$, $a_1=\frac{1}{3}$, $a_{n-1} - a_n = 2a_{n-1} \cdot a_n$ (where $n \geq 2$ and $n \in \mathbb{N}^*$), find the expression for $a_n$ and the sum $a_1a_2 + a_2a_3 + ... + a_na_{n+1}$.

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