
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
To stabilize housing prices, a local government decided to build a batch of affordable housing for the community. The plan is to purchase a piece of land for 16 million yuan and build a residential community with 10 buildings on it. Each building has the same number of floors, and each floor has a construction area of $1,000 m^{2}$. The construction cost per square meter is related to the floor level, with the cost for the $x^{th}$ floor being $(kx+800)$ yuan (where $k$ is a constant). It has been calculated that if each building has 5 floors, the average comprehensive cost per square meter for the community is 1,270 yuan.

Note: The average comprehensive cost per square meter $= \dfrac{\text{land purchase cost} + \text{all construction costs}}{\text{total construction area}}$.

$(1)$ Find the value of $k$;

$(2)$ To minimize the average comprehensive cost per square meter for the community, how many floors should the 10 buildings have? What is the average comprehensive cost per square meter at this time?

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