
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
A line $l$ passing through the point $(0,3b)$ is parallel to one of the asymptotes with a positive slope of the hyperbola $C$: $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$. If the distance from any point on the right branch of hyperbola $C$ to line $l$ is always greater than $b$, find the maximum value of the eccentricity of hyperbola $C$.

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