
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
For any positive integer  $n$ , we define the function  $f(n)$  as   $$ f(n)=\left\{\begin{aligned}
&n^{100} &&&& \text{if the sum of the digits of}~n~\text{is even}  
&-n^{100} &&&& \text{if the sum of the digits of}~n~\text{is odd} 
\end{aligned}\right. $$  Suppose  $S=f(1)+f(2)+\ldots+f(10^{100}-1)$ . Find the largest non-negative integer  $m$ , such that  $5^m\mid S$ . 

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