
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 $p:(x-a)(x-4a) \lt 0$ $(a \gt 0, x \in \mathbb{R})$, $q: 3 \lt x \lt 4$, $x \in \mathbb{R}$.  
$(Ⅰ)$ If $a=1$, determine the condition under which $q$ is a condition for $p$.  
$(Ⅱ)$ If $p$ is a necessary but not sufficient condition for $q$, find the range of real numbers for $a$.

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