
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 the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, let $O$, $F$, $A$, and $H$ denote its center, right focus, right vertex, and the intersection point of its right directrix with the $x$-axis, respectively. Determine the maximum value of $\frac{|FA|}{|OH|}$.

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