
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 \( f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 \) where \( a_0 \), \( a_1 \), \( a_2 \), \( a_3 \), and \( a_4 \) are constants with \( a_4 \neq 0 \). When divided by \( x - 2003 \), \( x - 2004 \), \( x - 2005 \), \( x - 2006 \), and \( x - 2007 \), respectively, \( f(x) \) leaves a remainder of 24, -6, 4, -6, and 24. Find the value of \( f(2008) \).

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