
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 county is creating a provincial-level sanitary and civilized county town, and is carrying out the rectification of the river channels in the county town. There is a 180-meter long river channel rectification task that is completed by two engineering teams, Team A and Team B, in succession. Team A rectifies 8 meters per day, Team B rectifies 12 meters per day, and they take a total of 20 days to complete the task.

$(1)$ Xiao Ming and Xiao Hua proposed the following solution ideas:<br/>Xiao Ming: Let the length of the river channel rectified by Team A be $x$ meters, and by Team B be $y$ meters.<br/>According to the question, we have $\left\{\begin{array}{l}{x+y=180}\\{\quad +\quad =20}\end{array}\right.$<br/>Xiao Hua: Let $m$ represent ______ after the completion of the rectification task, and let $n$ represent ______;<br/>We have $\left\{\begin{array}{l}{m+n=20}\\{\quad +\quad =\quad}\end{array}\right.$<br/>Please complete the solution ideas of Xiao Ming and Xiao Hua.

$(2)$ Find out how many meters of the river channel were rectified by Team A and Team B respectively? Please choose one set of equations to solve. (Write out the complete solution process)

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