
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 curves \( C_{1}: \frac{x^{2}}{a^{2}}+y^{2}=1 \) (where \( a \) is a constant) and \( C_{2}: y^{2}=2(x+m) \), such that they have only 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 let \( C_{1} \) intersect the negative part of the x-axis at point \( A \). When \( 0<a<\frac{1}{2} \), find the maximum value of the area of triangle \( \triangle OAP \) (in terms of \( a \)).

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