
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
Given a positive number \( p \) and the parabola \( C: y^{2} = 2 p x \) with \( p > 0 \), let \( A\left(\frac{p}{6}, 0\right) \) be a point on the axis of symmetry of the parabola \( C \), and \( O \) be the vertex of the parabola \( C \). Let \( M \) be any point on the parabola \( C \). Find the maximum value of \( \frac{|OM|}{|AM|} \).

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