
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
\[(ae)^2 + (bf)^2 + (cg)^2 + (dh)^2 + (ab)^2 + (cd)^2 + (ef)^2 + (gh)^2.\]

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