
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
Given the algebraic expression $x^2+4x$ can be transformed into $(x+2)^2-4$ using the formula for completing the square, it is known that the minimum value of $x^2+4x$ is $-4$. Following this method, the minimum value of the algebraic expression $x^2+y^2+6x-2y+12$ is ____.

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