
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
A chemical factory aims to predict the recycling rate ($y$) of a certain product by investigating the correlation between it and the effective component content of raw materials ($x$). They have obtained 8 sets of observation values and calculated the following:

$$\sum\limits_{i=1}^{8}{{{x}\_{i}}=52,\sum\limits_{i=1}^{8}{{{y}\_{i}}=228,\sum\limits_{i=1}^{8}{x\_{i}^{2}=478}}}$$, $$\sum\limits_{i=1}^{8}{{{x}\_{i}}{{y}\_{i}}=1849}$$

Determine the regression line equation for $y$ with respect to $x$.

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