
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 \( C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) \( (a>b>0) \) and the line \( x=-\sqrt{2} b \) which intersects the ellipse at exactly one point. Point \( P \) is any point on the ellipse \( C \), with \( P_{1}(-1,0) \) and \( P_{2}(1,0) \). The minimum value of \( \overrightarrow{P P_{1}} \cdot \overrightarrow{P P_{2}} \) is \( \frac{a}{2} \).

(1) Find the equation of the ellipse \( C \).

(2) Let the line \( l: y=k x+m \) intersect the ellipse \( C \) at points \( A \) and \( B \). Let \( O \) be the origin, and let \( \overrightarrow{O M}=\frac{1}{2}(\overrightarrow{O A}+\overrightarrow{O B}) \). When the area of triangle \( \triangle A O B \) is maximized, determine the range of the value of \( T=\frac{1}{\left|M P_{1}\right|^{2}}-2\left|M P_{2}\right| \).

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