
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
Define functions \( f \) and \( g \) such that for each non-zero real number \( x \) in their respective domains, \( 1/x \) is also in their domains and:
\[ f(x) + g\left(\frac{1}{x}\right) = x \]
\[ g(x) + f\left(\frac{1}{x}\right) = \frac{1}{x} \]
Determine \( f(1) \) and \( g(1) \).

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