
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
Call a simple graph  $G$  *quasicolorable* if we can color each edge blue, red, green, or white such that


- for each vertex v of degree 3 in G, the three edges incident to v are either (1) red,
green, and blue, or (2) all white,
- not all edges are white.

A simple connected graph  $G$  has  $a$  vertices of degree  $4$ ,  $b$  vertices of degree  $3$ , and no other vertices, where  $a$  and  $b$  are positive integers. Find the smallest real number  $c$  so that the following statement is true: “If  $a/b > c$ , then  $G$  must be quasicolorable.”

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