
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 the equation about $x$: $b(x^{2}-1)+2ax+c(x^{2}+1)=0$, where $a$, $b$, and $c$ are the lengths of the sides of triangle $\triangle ABC$.<br/>$(1)$ If $x=-1$ is a root of the equation, determine the shape of $\triangle ABC$;<br/>$(2)$ If the equation has two equal real roots, and $a=5$, $b=12$, find the value of $c$.

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