
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 a quadratic function in terms of $x$, $y = 4x^2 - 8nx - 3n - 2$, the difference in the $x$-coordinates of the two points where the graph of the quadratic function intersects the $x$-axis, squared, is equal to an integer root of the equation $x^2 - (7n + 6)x + 2(n + 1)(5n + 4) = 0$. Find the value of $n$.

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