
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
(1) Given $p$: $|x-a| < 4$, $q$: $-x^2 + 5x - 6 > 0$, and $q$ is a sufficient but not necessary condition for $p$, the range of values for $a$ is ______.

(2) If the proposition $p$: "There exists $x_0 \in \mathbb{R}$, $2^{x_0} - 2 \leqslant a^2 - 3a$" is a false proposition, then the range of values for the real number $a$ is ______.

(3) The proposition "For all $x \in \mathbb{R}$, $ax^2 - 2ax + 5 > 0$ always holds" is a false proposition, then the range of values for the real number $a$ is ______.

(4) Among the following propositions, the correct proposition numbers are ______.  
① The solution set of the system of equations $\begin{cases} 2x + y = 0 \\ x - y = 3 \end{cases}$ is $\{1, 2\}$,  
② The set $C = \left\{\frac{6}{3-x} \in \mathbb{Z} | x \in \mathbb{N}^*\right\} = \{-6, -3, -2, -1, 3, 6\}$,  
③ $f(x) = \sqrt{x-3} + \sqrt{2-x}$ is a function,  
④ $f(x) = ax^2 + bx + 3a + b$ is an even function, with domain $[a-1, 2a]$, then $f(0) = 1$,  
⑤ The number of sets $S$ that satisfy $S \subseteq A$ and $S \cap B \neq \varnothing$ for sets $A = \{1, 2, 3, 4\}$, $B = \{3, 4, 5, 6\}$ is 12,  
⑥ The function $y = \frac{2}{x}$ is a decreasing function in its domain.

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