
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 factorizing, such as grouping decomposition method, term splitting method, and so on.  
① Grouping decomposition method:  
For example, $x^{2}-2xy+y^{2}-4=(x^{2}-2xy+y^{2})-4=(x-y)^{2}-2^{2}=(x-y-2)(x-y+2)$.  
② Term splitting method:  
For example, $x^{2}+2x-3=x^{2}+2x+1-4=(x+1)^{2}-2^{2}=(x+1-2)(x+1+2)=(x-1)(x+3)$.  
$(1)$ Following the above methods, factorize as required:  
① (Grouping decomposition method) $4x^{2}+4x-y^{2}+1$;  
② (Term splitting method) $x^{2}-6x+8$;  
$(2)$ Given: $a$, $b$, $c$ are the three sides of $\triangle ABC$, $a^{2}+b^{2}+c^{2}-4a-4b-6c+17=0$, find the perimeter of $\triangle ABC$.

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