
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 $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{3}=1$ with $a > \sqrt {3}$, the center, left focus, left vertex, and the intersection point of the left directrix with the x-axis are denoted by O, F, G, H respectively. Find the value of $a$ when the ratio $\frac {FG}{OH}$ attains its maximum value.

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