
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 train station is continuously under construction. Currently, the station plans to build a storage room on one side of a warehouse, utilizing the existing wall, with a height of $3$ meters and a base area of $12$ square meters. The storage room will have a rectangular shape with one side against the wall. Since there is no need to build a wall on the back side against the existing wall, the construction team A offers the following pricing: $400$ yuan per square meter for building a new wall in front, $150$ yuan per square meter for building new walls on the left and right sides, and a total of $7200$ yuan for the roof, floor, and other expenses. The lengths of the left and right walls of the room are both $x$ meters $\left(2\leqslant x\leqslant 6\right)$.

$(1)$ When the lengths of the left and right walls are how many meters, will the construction team A offer the lowest price?

$(2)$ Another construction team B also participates in the bidding for the construction of this storage room. Their overall bid is $\frac{{900a(1+x)}}{x}$ yuan $\left(a \gt 0\right)$. If regardless of the lengths of the left and right walls, team B can successfully bid, find the range of values for $a$.

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