
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
Definition: If two quadratic surds $a$ and $b$ satisfy $a \cdot b = c$, and $c$ is a rational number, then $a$ and $b$ are called conjugate quadratic surds with respect to $c$.  
$(1)$ If $a$ and $\sqrt{2}$ are conjugate quadratic surds with respect to $4$, then $a =$ ______;  
$(2)$ If $3+\sqrt{3}$ and $6+\sqrt{3}m$ are conjugate quadratic surds with respect to $12$, find the value of $m$.

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