
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

At $12:00$, it follows that the paths intersect at an angle of $60^\circ$. Since they crossed the intersection afterwards, they are moving toward it. Since the cyclist took twice as much time on this road, his speed is half that of the motorcyclist, which is 36 km/h.

If time $t$ is measured starting from 13:00 (and negative values of $t$ are possible), then the motorcyclist at time $t$ is at a distance $|72t|$ from the intersection point, and the cyclist is at a distance $|36(t-1)|$. Therefore, by the cosine theorem:

$$
72^2 t^2 + 36^2 (t-1)^2 - 2 \cdot 72 \cdot 36 |t| |t-1| \cdot \frac{1}{2} = 252^2
$$

or after simplification:

$$
2^2 t^2 + (t-1)^2 - 2|t||t-1| = 7^2
$$

Obviously, values of $t$ between 0 and 1 do not fit (follows from a simple estimation of the distance between them), hence: $4t^2 + t^2 - 2t + 1 - 2t^2 + 2t = 49$, i.e., $t^2 = 16$. Therefore, the necessary times are: 09:00 and 17:00.

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