
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 ${a_{n}}$ where $a_{n} \neq 0$, $a_{1}= \frac {1}{3}$, and $a_{n-1}-a_{n}=2a_{n-1} \cdot a_{n}$ for $n\geqslant 2,n\in \mathbb{N}^{*}$, find $a_{n}$ and $a_{1}a_{2}+a_{2}a_{3}+…+a_{n}a_{n+1}$.

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