
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 school is planning to build a rectangular sports hall with a ground area of $240m^{2}$ and a height of $5m$. The first construction team, Team A, provides a quote: the cost of the roof per square meter is $100, the average cost of the walls on the front and back sides of the hall is $150 per square meter, and the average cost of the walls on the left and right sides is $250 per square meter. Let the length of the front wall of the sports hall be $x$ meters.  
$(1)$ When the length of the front wall is how many meters, is Team A's quote the lowest?  
$(2)$ Another construction team, Team B, also participates in the bidding for the construction of the sports hall at the school. Their overall quote is $12000+500({\frac{{a+1152}}{x}+a})$ yuan $\left(a \gt 0\right)$. If regardless of the length of the front wall, 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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