
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 equations \(a = \lg z + \lg \left[ x(yz)^{-1} + 1 \right]\), \(b = \lg x^{-1} + \lg \left( xyz + 1 \right)\), and \(c = \lg y + \lg \left[ (xyz)^{-1} + 1 \right]\), let \(M\) be the largest value among \(a\), \(b\), and \(c\). What is the minimum value of \(M\)?

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