
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 certain factory is building new dormitories for its employees. The relationship between the total cost $P (\text{in 10,000 yuan})$ of building the dormitory and the distance $x (\text{in km})$ between the dormitory and the factory is given by $P=\frac{k}{{3x+2}}$ for $0≤x≤5$. If the distance is $1$ km, the cost of building the dormitory is measured to be $40$ thousand yuan. For convenient transportation, a road needs to be built between the factory and the dormitory, with a known cost of $6$ thousand yuan per km. Let $y$ be the sum of the cost of building the dormitory and the cost of constructing the road.  
$(1)$ Find the value of $k$;  
$(2)$ Find the expression of $y$ in terms of $x$;  
$(3)$ Determine the distance from the factory where the dormitory should be built to minimize the total cost $y$, and find the minimum value.

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