
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
For every  $n$  in the set  $\mathrm{N} = \{1,2,\dots \}$  of positive integers, let  $r_n$  be the minimum value of  $|c-d\sqrt{3}|$  for all nonnegative integers  $c$  and  $d$  with  $c+d=n$ . Find, with proof, the smallest positive real number  $g$  with  $r_n \leq g$  for all  $n \in \mathbb{N}$ .

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