
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
The equation of the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ shares the same foci with the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$. If a line passing through the right focus $F$ with a slope of $60^\circ$ intersects the right branch of the hyperbola at two distinct points, find the range of values for the length of the real semi-axis of this hyperbola.

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