
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
Let [$a$ $b$] denote the average of $a^2$ and $b^2$, and {$a$ $b$ $c$} represent the average of $a$, $b$, and $2c$. Determine the value of $\{\{{2, -1, 1}\}, [2, 3], 1\}$.
**A)** $\frac{17}{6} \quad$ **B)** $\frac{19}{6} \quad$ **C)** $\frac{20}{6} \quad$ **D)** $\frac{22}{6} \quad$ **E)** $\frac{24}{6}$

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