
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 $2a_1+2^2a_2+2^3a_3+…+2^na_n=n$ (where $n\in\mathbb{N}^*$), and the sum of the first $n$ terms of the sequence $\left\{\frac{1}{\log_2{a_n}\cdot\log_2{a_{n+1}}}\right\}$ is denoted as $S_n$, find the product of $S_1\cdot S_2\cdot S_3\cdots S_{10}$.

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