
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 \( x, y, z \) be real numbers satisfying
\[ 
\frac{1}{x} + y + z = x + \frac{1}{y} + z = x + y + \frac{1}{z} = 3. 
\]
The sum of all possible values of \( x + y + z \) can be written as \( \frac{m}{n} \), where \( m \) and \( n \) are positive integers and \( \operatorname{gcd}(m, n) = 1 \). Find \( 100m + n \).

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