
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 unit is constructing a small room that occupies a ground area of $36m^{2}$ and is backed against a wall. Due to geographical limitations, the length $x$ of the small room's side cannot exceed $7m$, and the wall height is $2m$. The construction cost for the small room's front is $40$ yuan$/{m^{2}}$, while the side construction cost is $20$ yuan$/{m^{2}}$. The ground and other expenses amount to $1800$ yuan.
(1) Express the total construction cost $y$ as a function of $x$, and write out the function's domain.
(2) What should the length of the side be to minimize the total construction cost? What is the minimum total cost?

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