
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  $n \geq 3$  be a positive integer. Find the smallest positive real  $k$ , satisfying the following condition: if  $G$  is a connected graph with  $n$  vertices and  $m$  edges, then it is always possible to delete at most  $k(m-\lfloor \frac{n} {2} \rfloor)$  edges, so that the resulting graph has a proper vertex coloring with two colors. 

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