
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
Let $p$: The real number $x$ satisfies $x^2 - 4ax + 3a^2 < 0 (a > 0)$, and $q$: The real number $x$ satisfies $x = \left( \frac{1}{2} \right)^{m-1}$, where $m \in (1,2)$.

$(1)$ If $a = \frac{1}{4}$ and $p \wedge q$ is true, find the range of the real number $x$.

$(2)$ If $q$ is a sufficient but not necessary condition for $p$, find the range of the real number $a$.

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