
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
Ellipse \(\Gamma: x^{2}+\frac{y^{2}}{4}=1\) (\(0 \leq x \leq 1, 0 \leq y \leq 2\)) in the first quadrant has an arc represented by \(\overparen{A B}\). Point \(P(x, y)\) lies on arc \(\overparen{A B}\). Using \(t(P)\) to denote the unit tangent vector to ellipse \(\Gamma\) at point \(P\) in the counterclockwise direction:

(1) Let \(f(P)= t(P) \cdot \overrightarrow{O P}\); write the explicit formula for the function \(f(P)=f(x)\).
(2) Find the maximum value of the function \(f(P)\), the coordinates of point \(P\) where the maximum occurs, and determine the range of the function \(f(P)=f(x)\).

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