
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 $x^2-(m+2)x+m-2=0$ has two real roots $x_1$ and $x_2$ with opposite signs, and $x_1 > |x_2| > 0$; The equation $mx^2+(n-2)x+m^2-3=0$ has two rational roots and the product of the roots equals 2. Find the value of the integer $n$.

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