
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 \( C_{2}: y^{2} = 2(x + m) \) intersecting above the x-axis at exactly one common point \( P \):
(1) Find the range of real numbers \( m \) in terms of \( a \).
(2) Let \( O \) be the origin, and let the curves \( C_{1} \) and the negative x-axis intersect at point \( A \). When \( 0 < a < \frac{1}{2} \), find the maximum area of the triangle \( \triangle OAP \) in terms of \( a \).

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