
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
Let $a,$ $b,$ $c$ be nonnegative real numbers. Define:
\begin{align*}
P &= \sqrt{a + 3} + \sqrt{b + 6} + \sqrt{c + 12}, \\
Q &= \sqrt{a + 2} + \sqrt{b + 2} + \sqrt{c + 2}.
\end{align*}
Find the minimum value of $P^2 - Q^2.$

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