
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 (n) $  be a function that fulfills the following properties: $\bullet$  For each natural   $ n $ ,   $ f (n) $  is an integer greater than or equal to  $ 0 $ . $\bullet$   $f (n) = 2010 $ , if  $ n $  ends in  $ 7 $ . For example,   $ f (137) = 2010 $ . $\bullet$  If  $ a $  is a divisor of  $ b $ , then:   $ f \left(\frac {b} {a} \right) = | f (b) -f (a) | $ .

Find  $ \displaystyle f (2009 ^ {2009 ^ {2009}}) $  and justify your answer.

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