
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
Observe the pattern of the following equations and answer the following questions:<br/>①$3^{2}-3^{1}=2\times 3^{1}$<br/>②$3^{3}-3^{2}=2\times 3^{2}$<br/>③$3^{4}-3^{3}=2\times 3^{3}$<br/>$\ldots $<br/>$(1)$ According to this pattern, the 4th equation is______; the nth equation is______; (expressed in algebraic form containing n, where n is a positive integer)<br/>$(2)$ According to this pattern, calculate:<br/>①$2\times 3^{1}+2\times 3^{2}+2\times 3^{3}+2\times 3^{4}+2\times 3^{5}$<br/>②$3^{1}+3^{2}+3^{3}+\ldots +3^{n}$

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