
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 the function
$$
f(x, y) = \sqrt{x^{2}+y^{2}-6y+9} + \sqrt{x^{2}+y^{2}+2\sqrt{3}x+3} + \sqrt{x^{2}+y^{2}-2\sqrt{3}x+3}
$$

Find the minimum value of \( f(x, y) \).

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