
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
During the Winter Olympics, Bing Dwen Dwen became a hot-selling product. A Bing Dwen Dwen production company plans to rent land to build a temporary warehouse to store goods in order to increase production. If the distance from the warehouse to the station is $x$ (unit: km), according to market research, it is known that the monthly land occupation fee $y_{1}$ (unit: ten thousand yuan) is inversely proportional to $x+1$, and the monthly inventory storage fee $y_{2}$ (unit: ten thousand yuan) is directly proportional to $3x+1$. If the warehouse is built at a distance of $5$ km from the station, then $y_{1}$ and $y_{2}$ are $8$ ten thousand yuan and $16$ ten thousand yuan respectively. Let the sum of these two costs be $\omega = y_{1} + y_{2}$.
- $(1)$ Find the analytical expression of $\omega$ in terms of $x$.
- $(2)$ At what distance from the station should this company build the warehouse in order to minimize the sum of the two costs? What is the minimum value?

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