
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 \( f(x) = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4} + a_{5} x^{5} \) be a polynomial in \( x \) where \( a_{0}, a_{1}, a_{2}, a_{3}, a_{4} \) are constants and \( a_{5} = 7 \). When divided by \( x-2004, x-2005, x-2006, x-2007 \) and \( x-2008 \), \( f(x) \) leaves a remainder of \( 72, -30, 32, -24 \) and 24 respectively. Find the value of \( f(2009) \).

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