
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, a rectangle is called standard if all of its sides are parallel to the $x$- and $y$-axes, and a set of points is called nice if no two of them have the same $x$- or $y$-coordinates. Bert chooses a nice set $B$ of 2016 points in the coordinate plane. Ernie then chooses a set $E$ of $n$ points such that $B \cup E$ is a nice set with $2016+n$ points. Bert 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 $B$ chosen by Bert, 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. Determine the minimum and maximum possible values of $f(B)$.

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