
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 positive real numbers \(a\), \(b\), and \(c\) that satisfy:
\[ x = \frac{a}{2b + 3c}, \; y = \frac{2b}{3c + a}, \; z = \frac{3c}{a + 2b} \]
Calculate the value of \(\frac{x}{1 + x} + \frac{y}{1 + y} + \frac{z}{1 + z}\).

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