
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
For any real numbers \( a \) and \( b \), the inequality \( \max \{|a+b|,|a-b|,|2006-b|\} \geq C \) always holds. Find the maximum value of the constant \( C \). (Note: \( \max \{x, y, z\} \) denotes the largest among \( x, y, \) and \( z \).)

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