
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 that the equation \(x^{2}+a x+b-3=0\) (where \(a\) and \(b\) are real numbers) has a real root in the interval \([1,2]\), find the minimum value of \(a^{2}+(b-4)^{2}\).

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