
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
In the sequence $\{a_n\}$, it is known that $a_n = \frac{n^2+n-1}{3}$ $(n\in\mathbb{N}^*)$.  
(1) Write out $a_{10}$, $a_{n+1}$, $a_n^2$;  
(2) Is $79\frac{2}{3}$ an element of this sequence? If so, which term is it?

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