
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  $Z$  denote the set of points in  $\mathbb{R}^{n}$  whose coordinates are  $0$  or  $1.$  (Thus  $Z$  has  $2^{n}$  elements, which are the vertices of a unit hypercube in  $\mathbb{R}^{n}$ .) Given a vector subspace  $V$  of  $\mathbb{R}^{n},$  let  $Z(V)$  denote the number of members of  $Z$  that lie in  $V.$  Let  $k$  be given,  $0\le k\le n.$  Find the maximum, over all vector subspaces  $V\subseteq\mathbb{R}^{n}$  of dimension  $k,$  of the number of points in  $V\cap Z.$ 

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