
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 have learned methods to factorize a polynomial, such as the method of common factors and the application of formulas. In fact, there are other methods for factorization, such as grouping method and splitting method.<br/>① Grouping method:<br/>For example, $x^{2}-2xy+y^{2}-25=(x^{2}-2xy+y^{2})-25=(x-y)^{2}-5^{2}=(x-y-5)(x-y+5)$.<br/>② Splitting method:<br/>For example, $x^{2}+2x-3=x^{2}+2x+1-4=(x+1)^{2}-2^{2}=(x+1-2)(x+1+2)=(x-1)(x+3)$.<br/>$(1)$ Following the above methods, factorize as required:<br/>① Using grouping method: $9x^{2}+6x-y^{2}+1$;<br/>② Using splitting method: $x^{2}-6x+8$;<br/>$(2)$ Given: $a$, $b$, $c$ are the three sides of $\triangle ABC$, $a^{2}+5b^{2}+c^{2}-4ab-6b-10c+34=0$, find the perimeter of $\triangle ABC$.

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