
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
In a Cartesian coordinate plane, call a rectangle  $standard$  if all of its sides are parallel to the  $x$ - and  $y$ - axes, and call a set of points  $nice$  if no two of them have the same  $x$ - or  $y$ - coordinate. First, Bert chooses a nice set  $B$  of  $2016$  points in the coordinate plane. To mess with Bert, Ernie then chooses a set  $E$  of  $n$  points in the coordinate plane such that  $B\cup E$  is a nice set with  $2016+n$  points. Bert returns and then miraculously notices that there does not exist a standard rectangle that contains at least two points in  $B$  and no points in  $E$  in its interior. For a given nice set  $B$  that Bert chooses, define  $f(B)$  as the smallest positive integer  $n$  such that Ernie can find a nice set  $E$  of size  $n$  with the aforementioned properties. Help Bert determine the minimum and maximum possible values of  $f(B)$ .

*Yannick Yao*

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