
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 set of observed data $(x_i, y_i)$ ($i=1$, $2$, $...$, $8$) for linearly correlated variables $x$ and $y$, the regression line equation is $\hat{y} = \frac{1}{3}x + a$. Additionally, $x_1 + x_2 + ... + x_8 = 6$, $y_1 + y_2 + ... + y_8 = 3$. Determine the value of the real number $a$.

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