
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 \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) that passes through the fixed point \(A(1,0)\), and whose foci are on the \(x\)-axis, with the ellipse intersecting the curve \(|y|=x\) at points \(B\) and \(C\). Now, taking \(A\) as the focus, a parabola passing through \(B\) and \(C\) and opening to the left has its vertex at \(M(m, 0)\). When the eccentricity of the ellipse satisfies \(\sqrt{\frac{2}{3}}<e<1\), find the range of the real number \(m\).

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