
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
We know that factoring and polynomial multiplication are inverse operations. In the practice of factoring, we will also encounter the following problems. Please answer based on the situation:
$(1)$ Given that $a$, $b$, $c$ are the three sides of triangle $\triangle ABC$ and satisfy $a^{2}+2b^{2}=2b\left(a+c\right)-c^{2}$. Determine the shape of $\triangle ABC$;
$(2)$ Two students were factoring a quadratic trinomial. One student mistakenly factored it as $3\left(x-1\right)\left(x+2\right)$ due to misreading the coefficient of the linear term, while the other student mistakenly factored it as $3\left(x+2\right)\left(x-3\right)$ due to misreading the constant term. Please find the original polynomial and factorize it.

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