
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 the function \( f(x) = \left\{ 
\begin{aligned} 
2x-2 & \quad \text{if } x < 3 \\
\sqrt{2x} & \quad \text{if } x \ge 3 
\end{aligned} \right. \)
has an inverse \( f^{-1} \). Find the value of \( f^{-1}(-4) + f^{-1}(-2) + \dots + f^{-1}(6) \).

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