
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
If \(a, b, c > 0\), find the smallest possible value of
\[
\left\lfloor \frac{a^2+b^2}{a+b} \right\rfloor + \left\lfloor \frac{b^2+c^2}{b+c} \right\rfloor + \left\lfloor \frac{c^2+a^2}{c+a} \right\rfloor.
\]
(Note that \(\lfloor{x}\rfloor\) denotes the greatest integer less than or equal to \(x\).)

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