
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 functions $f(x) = \frac{1}{2}x^2$ and $g(x) = a\ln x$,
1. If the equation of the tangent line to the curve $y = f(x) - g(x)$ at $x = 1$ is $6x - 2y - 5 = 0$, find the value of the real number $a$;
2. Let $h(x) = f(x) + g(x)$, if for any two different positive numbers $x_1$, $x_2$, it always holds that $\frac{h(x_1) - h(x_2)}{x_1 - x_2} > 2$, find the range of the real number $a$;
3. If there exists a point $x_0$ in the interval $[1, e]$ such that $f'(x_0) + \frac{1}{f'(x_0)} < g(x_0) - g'(x_0)$, find the range of the real number $a$.

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