
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 company has two production lines, $A$ and $B$, processing the same raw materials. In one day, production line $A$ processes $a$ tons of raw materials in $\left(4a+1\right)$ hours; production line $B$ processes $b$ tons of raw materials in $\left(2b+3\right)$ hours.<br/>$(1)$ When $a=b=1$, how many hours does each production line take to process the materials?<br/>$(2)$ On the first day, the company allocated $5$ tons of raw materials to production lines $A$ and $B$, both lines completed the processing within a day, and the processing time was the same. What is the distribution of tons to the two production lines?<br/>$(3)$ Before starting work on the second day, the company allocated $5$ tons of raw materials based on the distribution results of the first day. Then, $A$ production line was allocated $m$ tons of materials, and $B$ production line was allocated $n$ tons of materials. If both production lines can process all the allocated raw materials within a day, and the processing time is the same, what is the relationship between $m$ and $n$? If the sum of $m$ and $n$ is $6$ tons at this time, what are the values of $m$ and $n$ in tons?

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