
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 a function $f(x)$ with domain $\mathbb{R}$ that has an inverse function $f^{-1}(x)$, and for any $x \in \mathbb{R}$, it always holds that $f(x) + f(-x) = 1$, determine the value of $f^{-1}(2010-x) + f^{-1}(x-2009)$.

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