
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 $\{a_{n}\}$, if ___.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ Find the sum of the first $n$ terms of the sequence $\{\frac{1}{{{a_n}{a_{n+1}}}}\}$.<br/>Choose one of the following conditions to fill in the blank above, then solve the problem.<br/>①$a_{1}+a_{2}+a_{3}+\ldots +a_{n}=n^{2}$;<br/>②$a_{1}=1$,$a_{4}=7$,$2a_{n}=a_{n-1}+a_{n+1}(n\in N*,n\geqslant 2)$;<br/>③$a_{1}=1$, point $A(n$,$a_{n})$,$B(n+1$,$a_{n+1})$ lie on a line with slope $2$.

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