
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
Read the following content before solving the problem: For a quadratic trinomial like $x^{2}+2ax+a^{2}$, it can be factored into the form of $(x+a)^{2}$ using the formula. However, for the quadratic trinomial $x^{2}+2ax-3a^{2}$, the formula cannot be directly applied. In this case, we can add $a^{2}$ to the quadratic trinomial $x^{2}+2ax-3a^{2}$, making it a perfect square with $x^{2}+2ax$, then subtract $a^{2}$. The value of the entire expression remains unchanged, so we have:  
$x^{2}+2ax-3a^{2}$  
$=(x^{2}+2ax+a^{2})-a^{2}-3a^{2}$  
$=\left(x+a\right)^{2}-4a^{2}$  
$=\left(x+a\right)^{2}-\left(2a\right)^{2}$  
$=\left(x+3a\right)\left(x-a\right)$  
This method of adding a suitable term to create a perfect square in the expression and then subtracting it to keep the value unchanged is called "completing the square." Use the "completing the square" method to solve the following problems:  
$(1)$ Factorize: $a^{2}-8a+15=$______；  
$(2)$ If the three sides of $\triangle ABC$ are $a$, $b$, $c$, and satisfy $a^{2}+b^{2}-14a-8b+65=0$, where the length of side $c$ is an odd number, find the minimum value of the perimeter of $\triangle ABC$;  
$(3)$ For what value of $x$ does the polynomial $-2x^{2}-4x+3$ have the maximum value? And find this maximum value.

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