
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 a line parameterized by
$$
\begin{cases}
x = t\cos\alpha + 3 \\
y = t\sin\alpha
\end{cases}
$$
where $t$ is a parameter, it is known that the line always passes through the right focus $F$ of the ellipse represented by
$$
\begin{cases}
x = 5\cos\theta \\
y = m\sin\theta
\end{cases}
$$
where $\theta$ is a parameter.
1. Find the eccentricity of the ellipse.
2. If the line intersects with the ellipse at points $M$ and $N$, find the maximum value of the product |MF| $\cdot$ |NF|.

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