
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 \( E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) with \(a > b > 0\), and its eccentricity \( e = \frac{\sqrt{3}}{2} \), the length of the semi-latus rectum (the chord perpendicular to the major axis and passing through a focus) is \( d = 1 \). Let \( F_{1} \) and \( F_{2} \) be the foci, and \( P \) be any point on the ellipse other than the endpoints of the major axis. The angle bisector of \(\angle F_{1} P F_{2} \), denoted as \( PM \), intersects the major axis at point \( M(m, 0) \). Determine the range of \( m \).

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