
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
When solving the equation system $\left\{\begin{array}{l}2ax+y=5\\ 2x-by=13\end{array}\right.$, due to carelessness, person A misread the variable $a$ in the equation system and obtained the solution $\left\{\begin{array}{l}x=\frac{7}{2}\\ y=-2\end{array}\right.$; person B misread the variable $b$ in the equation system and obtained the solution $\left\{\begin{array}{l}x=3\\ y=-7\end{array}\right.$.<br/>$(1)$ What did person A mistakenly read $a$ as? What did person B mistakenly read $b$ as?<br/>$(2)$ Find the correct solution to the original equation system.

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