
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
Consider the curves \( C_1: \frac{x^2}{a^2} + y^2 = 1 \) (where \( a \) is a constant) and \( C_2: y^2 = 2(x+m) \) which have exactly one common point \( P \) above the x-axis.
1. Find the range of the real number \( m \) in terms of \( a \).
2. Let \( O \) be the origin and \( C_1 \) intersects the negative x-axis at point \( A \). For \( 0 < a < \frac{1}{2} \), find the maximum value of the area of triangle \( \triangle OAP \).

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