
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 that is symmetric with respect to both coordinate axes, with its foci located on the \( y \)-axis and the length of its transverse axis being \( 2 \sin \theta \) where \( \theta \in \left[ \frac{\pi}{4}, \frac{\pi}{3} \right] \). Furthermore, for any point \( P(x, y) \) on the hyperbola, the shortest distance to the point \( M(1,0) \) is \( \frac{1}{\sin \theta} \). Determine the range of values for the eccentricity of the hyperbola.

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