
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 that $A$, $B$, and $C$ are all polynomials, where $A=x^{2}+xy+3y$ and $B=x^{2}-xy$.
$(1)$ Find the value of $3A-B$;
$(2)$ Lily made a mistake when calculating $A+\frac{1}{3}C$ and obtained $A-\frac{1}{3}C$, resulting in $2x^{2}+y$. Help Lily find the correct result.

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