
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 that point \( A(0,2) \) is a point outside the circle \( M: x^{2} + y^{2} - 2ax - 2ay = 0 \) (where \( a > 0 \)), there exists a point \( T \) on the circle \( M \) such that \( \angle MAT = 45^{\circ} \). Determine the range of the real number \( a \).

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