
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
Note that there are exactly three ways to write the integer  $4$  as a sum of positive odd integers where the order of the summands matters:
\begin{align*}
1+1+1+1&=4,
1+3&=4,
3+1&=4.
\end{align*}
Let  $f(n)$  be the number of ways to write a natural number  $n$  as a sum of positive odd integers where the order of the summands matters.  Find the remainder when  $f(2008)$  is divided by  $100$ .

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