
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
To factorize the quadratic trinomial $x^{2}+4x-5$, we can first add $4$ to $x^{2}+4x$ to make it a perfect square trinomial. Then, subtract $4$ so that the value of the entire expression remains unchanged. Therefore, we have: $x^{2}+4x-5=x^{2}+4x+4-4-5=\left(x+2\right)^{2}-9=\left(x+2+3\right)\left(x+2-3\right)=\left(x+5\right)\left(x-1\right)$. This method of first adding a suitable term to create a perfect square trinomial and then subtracting that term to keep the value of the entire expression unchanged is called "completing the square."

Please use the "completing the square" method to solve the following problems:
1. Factorize the expression: $x^{2}-120x+3456$.
2. Given $x^{2}+y^{2}+8x-12y+52=0$, find the value of $xy$.

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