
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 $\alpha$ and $\beta$ be real numbers. Find the minimum value of
\[
(3 \cos \alpha + 6 \sin \beta - 10)^2 + (3 \sin \alpha + 6 \cos \beta + 4\cos(\alpha + \beta) - 20)^2.
\]

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