
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
To implement the development concept of "green waters and green mountains are golden and silver mountains", a municipal department in a city invited bids for a construction team to be responsible for the earthwork construction of a reservoir at the foot of a mountain. The construction team has two types of excavators, $A$ and $B$. It is known that when $3$ type $A$ and $5$ type $B$ excavators work together for one hour, they can excavate $165$ cubic meters of soil; when $4$ type $A$ and $7$ type $B$ excavators work together for one hour, they can excavate $225$ cubic meters of soil. The construction cost per hour for each type $A$ excavator is $300$ yuan, and for each type $B$ excavator is $180$ yuan. <br/>$(1)$ Find out how many cubic meters of soil each type $A$ and $B$ excavator can excavate per hour, respectively. <br/>$(2)$ If different numbers of type $A$ and $B$ excavators, totaling $12$, work together for $4$ hours, and at least $1080$ cubic meters of soil need to be excavated, with the total cost not exceeding $12960$ yuan. Determine the possible allocation schemes during construction, and identify which allocation scheme has the lowest construction cost and what is the lowest cost in yuan?

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