
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
Assume $x$ and $y$ are positive numbers, and that $x, a_1, a_2, y$ form an arithmetic sequence, while $x, b_1, b_2, y$ form a geometric sequence. Find the minimum value of $\left(\frac{a_1 + a_2}{\sqrt{b_1 b_2}}\right)^2$.

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