
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
Among the following five propositions about conic sections:
(1) The equation of the trajectory of points in a plane with a distance ratio of $\frac{1}{2}$ to a fixed point $A(1, 0)$ and a fixed line $l: x = 2$ is $\frac{x^2}{4} + \frac{y^2}{3} = 1$.
(2) Let point $P$ be a moving point on the parabola $y^2 = 2x$, and the projection of point $P$ on the y-axis is point $M$. The coordinates of point $A$ are $A(3, 6)$, then the minimum value of $|PA| + |PM|$ is 6.
(3) The trajectory of points in a plane whose distances to two fixed points have a constant ratio $\lambda$ ($\lambda > 0$) is a circle.
(4) If a moving point $M(x, y)$ satisfies $\sqrt{(x-1)^2 + (y+2)^2} = |2x - y - 4|$, then the trajectory of point $M$ is a hyperbola.
(5) If a line $l$ passing through point $C(1, 1)$ intersects the ellipse $\frac{x^2}{4} + \frac{y^2}{3} = 1$ at two distinct points $A$ and $B$, and $C$ is the midpoint of $AB$, then the equation of line $l$ is $3x + 4y - 7 = 0$.
Indicate the correct propositions with their index numbers.

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