
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 $a, b, c, d, e, f, g, h$ be real numbers such that $abcd = 8$ and $efgh = 16$. Find the minimum value of 
\[
(2ae)^2 + (2bf)^2 + (2cg)^2 + (2dh)^2.
\]

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