
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
The ellipse \(\Gamma: x^{2}+\frac{y^{2}}{4}=1\) \((0 \leqslant x \leqslant 1, 0 \leqslant y \leqslant 2)\) in the first quadrant has an arc denoted as \(\overparen{A B}\). Point \(P(x, y)\) lies on the arc \(\overparen{A B}\). Let \(t(P)\) represent the unit tangent vector to the ellipse \(\Gamma\) at point \(P\), oriented counterclockwise along the ellipse.

(1) Let \(f(P) = t(P) \cdot \overrightarrow{O P}\). Write the analytical expression for the function \(f(P) = f(x)\).

(2) Find the maximum value of the function \(f(P)\) and the coordinates of point \(P\) where this maximum occurs. Determine the range of values for the function \(f(P) = f(x)\).

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