
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 the operation "$\Phi$" defined as: 
when $m < n$, $m\Phi n = 2\sqrt{m} + \sqrt{n}$; when $m \geqslant n$, $m\Phi n = 2\sqrt{m} - \sqrt{n}$, calculate the result of $\left(3\Phi 2\right) - \left(8\Phi 12\right)$.

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