
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 hyperbola $\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1$ where $a>0$ and $b>0$, the right focus is $F(4,0)$. If the line that passes through point F and has an inclination of $60^\circ$ intersects with only one point on the right branch of the hyperbola, determine the range of values for $a$.

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