
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
Suppose the statistical data related to the service life $x$ (unit: years) and the maintenance cost $y$ (unit: ten thousand yuan) of a certain device are as shown in the table below:

| Service Life $x$ (years) | $2$ | $3$ | $4$ | $5$ | $6$ |
|--------------------------|-----|-----|-----|-----|-----|
| Maintenance Cost $y$ (ten thousand yuan) | $2.2$ | $3.8$ | $5.5$ | $6.5$ | $7$ |

$(1)$ Find the linear regression equation $\hat{y}=\hat{b}x+\hat{a}$;  
$(2)$ Estimate the maintenance cost when the service life is $10$ years.

$\hat{b}=\frac{\sum_{i=1}^{n}({x}_{i}-\overline{x})({y}_{i}-\overline{y})}{\sum_{i=1}^{n}({x}_{i}-\overline{x})}=\frac{\sum_{i=1}^{n}{x}_{i}{y}_{i}-n\overline{xy}}{\sum_{i=1}^{n}{x}_{i}^{2}-n{\overline{x}}^{2}$.

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