
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
The advertising expenses $x$ (in units of 10,000 yuan) and sales revenue $y$ (in units of 10,000 yuan) of a certain product correspond to the data in the table below:

| $x$ | $1$ | $3$ | $4$ | $5$ | $7$ |
|-----|-----|-----|-----|-----|-----|
| $y$ | $15$ | $20$ | $30$ | $40$ | $45$ |

Based on the data in the table, the empirical regression equation of $y$ with respect to $x$ is $\hat{y}=4.5x+a$. Find the value of $a$.

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