
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
Consider the equation  $ x^4 \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} 2007$ , where  $ a,b,c$  are real numbers. Determine the largest value of  $ b$  for which this equation has exactly three distinct solutions, all of which are integers.

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