
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 popularity of electric cars is increasing, and they can be charged through fixed charging stations. A certain shopping mall plans to install public charging stations in the underground parking lot to meet the needs of customers. According to market analysis, the total profit $y$ (in units of ten thousand yuan) of the public charging stations is a quadratic function of the number of years of operation $x$ (where $x$ is a positive integer). The total profit is $20$ thousand yuan after three years of operation, and the profit is maximized at $110$ thousand yuan after six years of operation.  
$(1)$ Find the functional relationship between $y$ and $x$.  
$(2)$ Find the maximum value of the average annual total profit (Note: the average annual total profit $=$ total profit $/$ number of years of operation).

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