
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
Consider $x^3 - 18x^2 + 32x - 12 = 0.$ Let $a,$ $b,$ and $c$ be the roots. Find the value of $\frac{a}{\sqrt{a^2}+bc} + \frac{b}{\sqrt{b^2}+ca} + \frac{c}{\sqrt{c^2}+ab}.$

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