
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 \( G \) quasi-colorable if we can color each edge blue, red, green, and white such that
- for each vertex \( v \) of degree 3 in \( G \), the three edges containing \( v \) as an endpoint are either colored blue, red, and green, or all three edges are white,
- not all edges are white.

A 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 \( \frac{a}{b} > c \), then \( G \) is quasi-colorable."

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