
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 the set $\{a\_1,a\_2,...,a\_n\}$ and the constant $a\_0$, define $w= \frac {\sin ^{2}(a\_1-a\_0)+\sin ^{2}(a\_2-a\_0)+...+\sin ^{2}(a\_n-a\_0)}{n}$ as the "sine variance" of the set relative to $a\_0$. Evaluate the "sine variance" of the set $\{\frac {\pi}{2}, \frac {5\pi}{6}, \frac {7\pi}{6}\}$ relative to $a\_0$.

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