
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 the number of times you have to hit the  $ \sqrt {\ }$  key on a calculator to get a number less than  $ 2$  starting from  $ n$ . For instance,  $ f(2) \equal{} 1$ ,  $ f(5) \equal{} 2$ . For how many  $ 1 < m < 2008$  is  $ f(m)$  odd?

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