
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
Sasha takes one step in 1 second, and Danya takes one step in $\frac{6}{7}$ seconds. In 6 seconds, Sasha will have taken 6 steps, and Danya will have taken 7 steps. Therefore, we need to consider moments of time that are multiples of 6 seconds, i.e., $t = 6k$ for $k=1, 2, \ldots$. Sasha's step length is 0.4 m, and Danya's step length is 0.6 m, so in $t=6k$ seconds, Sasha will cover $6 \cdot k \cdot 0.4=2.4k$ meters, and Danya will cover $7 \cdot k \cdot 0.6=4.2k$ meters. The distance between Sasha and Danya at these moments in time is:

$$
d(k)=|200-2.4k| + |300-4.2k|.
$$

Consider the function $d(x)$ for $x \in \mathbb{R}$:
- For $x \leq x_{1}=\frac{300}{4.2}\approx 71.42$:
$$
d(x)=200-2.4x + 300-4.2x = 500-6.6x.
$$
- For $x_{1} \leq x \leq x_{2} = \frac{200}{2.4}\approx 83.33$:
$$
d(x)=200-2.4x - 300 + 4.2x = 1.8x-100.
$$
- For $x \geq x_{2}$:
$$
d(x)=-200+2.4x - 300 + 4.2x = 6.6x-500.
$$

The function $d(x)$ decreases for $x \leq x_{1}$ and increases for $x \geq x_{1}$, indicating that the function reaches its minimum at $x = x_{1}$. To find the minimum value of $d(k)$, consider the nearest integer values $x_{1}$: $k_1 = \lfloor x_{1} \rfloor = 71$ and $k_2 = \lfloor x_{1} \rfloor + 1 = 72$. Calculating, we find $d(71) = 500 - 6.6 \cdot 71 = 31.4$ and $d(72) = 1.8 \cdot 72 - 100 = 29.6$. Therefore, the smallest possible distance between Sasha and Danya is 29.6 m. Given this minimum occurs at $k=72$, the number of steps taken by Sasha by this time is $6 \cdot 72 = 432$, and the number of steps taken by Danya is $7 \cdot 72 = 504$.

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