
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 each real number \( x \), let \( f(x) \) be the minimum of the numbers \( 4x+1 \), \( x+2 \), and \( -2x+4 \). Determine the maximum value of \( 6 f(x) + 2012 \).

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