
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 $\{a_{n}\}$ that satisfies $a_{1}+3a_{2}+\cdots +\left(2n-1\right)a_{n}=n$.<br/>$(1)$ Find the general formula for $\{a_{n}\}$;<br/>$(2)$ Given $c_{n}=\left\{\begin{array}{l}{\frac{1}{19{a}_{n}}, \text{if } n \text{ is odd}}\\{{a}_{n}{a}_{n+2}, \text{if } n \text{ is even}}\end{array}\right.$, find the sum of the first $20$ terms of the sequence $\{c_{n}\}$.

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