
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
Define a permutation of the set  $\{1,2,3,...,n\}$  to be  $\textit{sortable}$  if upon cancelling an appropriate term of such permutation, the remaining  $n-1$  terms are in increasing order. If  $f(n)$  is the number of sortable permutations of  $\{1,2,3,...,n\}$ , find the remainder when   $$ \sum\limits_{i=1}^{2018} (-1)^i \cdot f(i) $$   is divided by  $1000$ . Note that the empty set is considered sortable.

*Proposed by **FedeX333X***

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