
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
Renowned mathematician Hu Luogeng once said: "The combination of mathematics and geometry is beneficial in a hundred ways, while separation and division bring nothing but stagnation." In fact, many algebraic problems can be transformed into geometric problems for solution. For example, $\sqrt{(x-a)^2 +(y-b)^2 }$ can be transformed into the distance between points $M(x, y)$ and $N(a, b)$ on a plane. Based on this perspective, find the minimum value of $f(x)=\sqrt{x^2 +4x+20}+ \sqrt{x^2 +2x+10}$.

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