
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 $f(x) = \log_{\frac{1}{2}} \left[ x^2 - 2(2a-1)x + 8 \right]$.
(1) If the domain of $f(x)$ is $\mathbb{R}$, find the range of values for $a$;
(2) If the range of $f(x)$ is $\mathbb{R}$, find the range of values for $a$;
(3) If $f(x)$ is defined on $[-1, +\infty]$, find the range of values for $a$;
(4) If $f(x)$ is decreasing on $[a, +\infty]$, find the range of values for $a$;
(5) If $a = \frac{3}{4}$, find the range of $y = f\left[ \sin(2x - \frac{\pi}{3}) \right]$ when $x \in \left[ \frac{\pi}{12}, \frac{\pi}{2} \right]$;
(6) If the equation $f(x) = -1 + \log_{\frac{1}{2}}(x+3)$ has exactly one solution in $[1, 3]$, find the value of $a$;
(7) Find the range of values for $a$ such that $f(x) \leqslant -1$ holds true for all $x \in [2, 3]$.

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