
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 function $f(x) = x^2 - a\ln x$:
1. If $f(x)$ is a monotonically decreasing function on the interval $[3, 5]$, determine the range of values for the real number $a$.
2. Let $g(x) = f(x) + (2+a)\ln x - 2(b-1)x$. Suppose $x_1$ and $x_2$ ($x_1 < x_2$) are the two extreme points of the function $g(x)$, and $b \geq \frac{7}{2}$, find the minimum value of $g(x_1) - g(x_2)$.

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