
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 \( C_{1}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) \((a > b > 0)\) and the hyperbola \( C_{2}: \frac{x^{2}}{m^{2}}-\frac{y^{2}}{n^{2}}=1 \) \((m > 0, n > 0)\) with the same foci \( F_{1} \,\text{and}\, F_{2} \), where \( F_{1} \) is the left focus. Point \( P \) is the intersection point of the two curves in the first quadrant. The eccentricities of the curves \( C_{1} \) and \( C_{2} \) are \( e_{1} \) and \( e_{2} \), respectively. If the triangle \( \triangle P F_{1} F_{2} \) is an isosceles triangle with \( P F_{1} \) as the base, find the range of the value \( e_{2} - e_{1} \).

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