
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 the sequence ${a_n}$ that satisfies the equation $2a_1+2^2a_2+2^3a_3+...+2^na_n=n (n\in N^*)$, and the sum of the first $n$ terms of the sequence ${\frac{1}{\log_2a_n \cdot \log_2a_{n+1}}}$ is $S_n$. Find the value of $S_1 \cdot S_2 \cdot S_3 ... S_{10}$.

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