
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
For obtaining photographs of celestial bodies, space probes—autonomous robots equipped with rocket engines, their own power systems, radio communication, navigation systems, and scientific instruments—are used. For instance, such probes have successfully executed the exploration program of Saturn and its largest moon, Titan. While studying one of Saturn's moons with an orbital radius \( R \approx 1.2 \cdot 10^5 \) km, an emergency situation occurred: during the probe's passage through Saturn's ring plane, the onboard rotating platform with cameras got jammed by particles from these rings. Consequently, clear images were obtained only from one side of the moon. To photograph the other side, it was decided to continue the probe's flight to meet the moon at another point in space, requiring a correction of the probe's speed.

Consider a simplified model of this scenario. The moon's trajectory (orbit) around Saturn (point \( O \)) is circular with a radius \( R = 1.2 \cdot 10^5 \) km, and the moon's constant velocity is \( V_T = 3.27 \) km/s. The projection of the probe onto the plane of the orbit will be called the sub-probe point. The sub-probe point's speed is constant and equals \( V_1 = 6 \) km/s, and its trajectory in the plane of the orbit is a straight line intersecting the orbit at points \( C \) and \( D \). According to the pre-programmed plan, the probe photographs the moon's surface at moments of closest approach, which corresponds to the intersections of the sub-probe point's trajectory with the moon's orbit (points \( C \) and \( D \)). When the moon (point \( S \)) is exactly on the line between the center of Saturn (point \( O \)) and the sub-probe point (point \( B \)), a timer starts (\( t_0 = 0 \)). The moon and the sub-probe point meet at point \( C \) after time \( t = 2 \cdot 10^4 \) s. After photographing at point \( C \), the probe's speed changes so that it meets the moon at point \( D \) simultaneously to photograph the other side. The speed of the sub-probe point on the section \( CD \) is constant.

Determine the distance between the sub-probe point and the moon (considering it a material point) at the initial moment \( t_0 \), as well as the speed \( V_2 \) of the sub-probe point on section \( CD \). Use approximate values for the moon's speed and \(\pi\)—round them to whole numbers.

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