
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
Suppose proposition $p$: a real number $x$ satisfies $(x - a)(x - 3a) < 0$, where $a > 0$. Proposition $q$: a real number $x$ satisfies $2 < x \leqslant 3$.
1. If $a = 1$, given that $p$ and $q$ are both true, determine the range of $x$.
2. If the negation of $p$ (denoted as $\neg p$) is a sufficient but not necessary condition for the negation of $q$ (denoted as $\neg q$), determine the range of the real number $a$.

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