
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
Find the largest  $r$  such that  $4$  balls each of radius  $r$  can be packed into a regular tetrahedron with side length  $1$ . In a packing, each ball lies outside every other ball, and every ball lies inside the boundaries of the tetrahedron. If  $r$  can be expressed in the form  $\frac{\sqrt{a}+b}{c}$  where  $a, b, c$  are integers such that  $\gcd(b, c) = 1$ , what is  $a + b + c$ ?


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