
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 a quadratic equation in terms of $x$: $\left(a+c\right)x^{2}+2bx+\left(b-c\right)=0$, where $a$, $b$, and $c$ are the three side lengths of $\triangle ABC$. <br/>$(1)$ If $x=-1$ is a root of the equation, determine the shape of $\triangle ABC$ and explain the reason. <br/>$(2)$ If $\triangle ABC$ is an equilateral triangle, find the roots of this quadratic equation.

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