
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  $S$  be the set of all partitions of  $2000$  (in a sum of positive integers). For every such partition  $p$ , we deﬁne  $f (p)$  to be the sum of the number of summands in  $p$  and the maximal summand in  $p$ . Compute the minimum of  $f (p)$  when  $p \in S .$ 

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