
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 $A= \begin{bmatrix} 1 & 0 \\ -1 & 1\end{bmatrix}$, $B= \begin{bmatrix} 1 & 2 \\ 0 & 3\end{bmatrix}$, and $C=AB$.
$(1)$ Find the matrix $C$;
$(2)$ If line $l_{1}$: $x+y=0$ is transformed by the matrix $C$ into another line $l_{2}$, find the equation of $l_{2}$.

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