
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
Given matrices $A=\left[ \begin{matrix} 1 & 0 \\ 0 & 2 \\ \end{matrix} \right]$, $B=\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \\ \end{matrix} \right]$, if the transformation corresponding to matrix $AB^{-1}$ turns a line into line $l':x+y-2=0$,

$(1)$ Find the matrix $B^{-1}$

$(2)$ Find the equation of the line.

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