
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
(1) Compare the size: $\sqrt{3} + \sqrt{5}$ ______ $\sqrt{2} + \sqrt{6}$ (Fill in the blank with ">" or "<")

(2) Convert the parametric equation of the curve $\begin{cases} x = 3 + \frac{1}{2}t \\ y = \frac{\sqrt{3}}{2}t \end{cases}$ (where $t$ is the parameter) into a regular equation ______.

(3) There are three cards, each marked with $1$, $2, 1$, and $3, 2$, and $3$. Person A, B, and C each take one card. After A sees B's card, A says: "The number that is the same on B's and my card is not $2$". After B sees C's card, B says: "The number that is the same on C's and my card is not $1$". C says: "The sum of the numbers on my card is not $5$". Then, the numbers on A's card are ______.

(4) The equation of the tangent line passing through a point $M(x_0, y_0)$ on the circle $x^2 + y^2 = r^2$ is $x_0x + y_0y = r^2$. By analogy, a similar property for the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is: the equation of the tangent line passing through a point $P(x_0, y_0)$ on the ellipse is ______.

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