
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
Find all values of the parameter \( a \) for which the system

$$
\left\{\begin{array}{c}
\left(a\left|\log _{2} y\right|+a\left|\log _{2} x\right|-2\right)\left(\left(\log _{2} x\right)^{2}+\left(\log _{2} y\right)^{2}-48 a^{2}\right)=0 \\
\left(\log _{a} x\right)^{2}\left(\log _{a} y\right)^{2}\left(\log _{2} a^{2}\right)^{4}=256 a^{2}
\end{array}\right.
$$

has eight distinct solutions.

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