
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
$(1)$ Observing a sequence of numbers $a_{1}=3$, $a_{2}=3^{2}$, $a_{3}=3^{3}$, $a_{4}=3^{4}$, $\ldots$, it is found that starting from the second term, the ratio of each term to the previous one is a constant, and this constant is ______; according to this rule, if $a_{n}$ (where $n$ is a positive integer) represents the $n$-th term of this sequence, then $a_{6}=$______, $a_{n}=\_\_\_\_\_\_$; (express in the form of powers)<br/>$(2)$ If we want to find the value of $1+2+2^{2}+2^{3}+\ldots +2^{10}$, we can let $S_{10}=1+2^{2}+2^{2}+2^{3}+\ldots +2^{10}$①<br/>Multiplying both sides of ① by $2$, we get ______②, subtracting ① from ②, we get $S_{10}=\_\_\_\_\_\_$.<br/>$(3)$ If the sequence in (1) has a total of $20$ terms, let $S_{20}=3+3^{2}+3^{3}+3^{4}+\ldots +3^{20}$, please calculate the value of $S_{20}$ using the above rule and method.

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