
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
$(1)$ Given the general formula of the sequence $\{a_{n}\}$ as $a_{n}=\frac{{2^n+1}}{{4^n}}$, find the sum of the first $n$ terms $S_{n}$ of $\{a_{n}\}$;<br/>$(2)$ Given the general formula of the sequence $\{a_{n}\}$ as $a_{n}=3n-1$, find the value of ${a}_{1}^{2}-{a}_{2}^{2}+{a}_{3}^{2}-{a}_{4}^{2}+\cdots +{a}_{19}^{2}-{a}_{20}^{2}$.

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