
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
Let the curves \(C_{1}: \frac{x^{2}}{a^{2}} + y^{2} = 1\) (where \(a\) is a positive constant) and \(C_{2}: y^{2} = 2(x + m)\) have only one common point \(P\) above the \(x\)-axis.
(1) Find the range of the real number \(m\) (denoted by \(\mu\)).
(2) Let \(O\) be the origin. If \(C_{1}\) intersects the negative half of the \(x\)-axis at point \(A\), and given \(0 < a < \frac{1}{2}\), find the maximum value of the area of \(\triangle OAP\) (expressed in terms of \(a\)).

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