
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 \( a = \lg z + \lg \left[ \frac{x}{yz} + 1 \right] \), \( b = \lg \frac{1}{x} + \lg (xyz + 1) \), and \( c = \lg y + \lg \left[ \frac{1}{xyz} + 1 \right] \). Denote the maximum of \( a \), \( b \), and \( c \) as \( M \). What is the minimum value of \( M \)?

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