
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 that the sequence $\{a_{n}\}$ is a non-zero sequence, and satisfies $({1+\frac{1}{{{a_1}}}})({1+\frac{1}{{{a_2}}}})…({1+\frac{1}{{{a_n}}}})={2^{\frac{{n({n+1})}}{2}}}$.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ Find the sum $S_{n}$ of the first $n$ terms of the sequence $\left\{{\frac{n}{{{a_n}}}}\right\}$.

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