
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 location plans to build a road, but the planned route passes through a wild animal migration route. In order to protect the wild animals, it is decided to build a viaduct to provide a safe passage for the migration of wild animals. If the two ends of the viaduct and the piers at both ends have been built, the distance between the piers at the two ends is 1200 meters, and the remaining work only needs to build the bridge deck and piers between the two end piers. According to predictions, the construction cost of one pier is 500 million yuan, and the construction cost of the bridge deck between adjacent piers at a distance of x meters is 10x[ln(x+12)-3] million yuan. Assuming that the piers are evenly distributed at equal distances, all piers are considered as points, and other factors are not considered. Let the cost of the remaining work be y million yuan.  
$(1)$ Write the function relationship of y with respect to x.  
$(2)$ How many piers need to be built to minimize y? And find its minimum value. Reference data: ln 2 ≈ 0.69, ln 3 ≈ 1.10.

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