
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]


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Given the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)\), the distance between the upper vertex \(A\) and the left vertex \(B\) is \(\sqrt{41}\), and the eccentricity is \(\frac{3}{5}\). Point \(P(t, 0)\) \((-4 \leqslant t \leqslant -1)\) is on the x-axis.
(1) Find the equation of the ellipse;
(2) Connect \(A\) and \(P\) which intersects the ellipse at point \(C\). Draw a perpendicular from point \(C\) to the x-axis, intersecting the ellipse at another point \(D\). Find the range of values for the area \( S_{\triangle A B D} \).

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