
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  $n>1$  be an integer. Given a simple graph  $G$  on  $n$  vertices  $v_1, v_2, \dots, v_n$  we let  $k(G)$  be the minimal value of  $k$  for which there exist  $n$   $k$ -dimensional rectangular boxes  $R_1, R_2, \dots, R_n$  in a  $k$ -dimensional coordinate system with edges parallel to the axes, so that for each  $1\leq i<j\leq n$ ,  $R_i$  and  $R_j$  intersect if and only if there is an edge between  $v_i$  and  $v_j$  in  $G$ .

Define  $M$  to be the maximal value of  $k(G)$  over all graphs on  $n$  vertices. Calculate  $M$  as a function of  $n$ .

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