
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 $\{a_{n}\}$ such that $a_{1}+2a_{2}+\cdots +na_{n}=n$, and a sequence $\{b_{n}\}$ such that ${b_{m-1}}+{b_m}=\frac{1}{{{a_m}}}({m∈N,m≥2})$. Find:<br/>
$(1)$ The general formula for $\{a_{n}\}$;<br/>
$(2)$ The sum of the first $20$ terms of $\{b_{n}\}$.

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