
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
Let \( x \) (km/h) be the speed of the pedestrian, \( y \) (km/h) be the initial speed of the cyclist, \( t \) (hours) be the time taken by the cyclist to travel from \( A \) to \( B \). Then, 

\[
\begin{cases} 
x(t + 0.2) + 0.4y = 12 \\
yt = 12 \\
t < 1 
\end{cases} \Rightarrow x\left(\frac{12}{y} + 0.2\right) + 0.4y = 12 \Rightarrow 2y^2 + (x - 60)y + 60x = 0
\]

For the quadratic equation to have solutions, it is necessary that \( D = (x - 60)^2 - 480x \geq 0 \). Consequently, \( x^2 - 600x + 3600 \geq 0 \), \( D / 4 = (120\sqrt{6})^2 \),

\[
x \in (-\infty; 300 - 120\sqrt{6}] \cup [300 + 120\sqrt{6}; +\infty)
\]

Since \( x \in \mathbb{N} \) and \( 0.2x < 12 \), i.e., \( x < 60 \), then \( x \in [1; 300 - 120\sqrt{6}] \cap \mathbb{N} \). Using the approximation \( 2.44 < \sqrt{6} < 2.45 \), we get \( 293 < 120\sqrt{6} < 294 \), and \( 6 < 300 - 120\sqrt{6} < 7 \). The greatest possible integer value of the speed \( x = 6 \). We'll find the initial speed of the cyclist for \( x=6 \) from the equation \( 2y^2 - 54y + 360 = 0 \), or \( y^2 - 27y + 180 = 0 \), \( y_1 = 12 \), \( y_2 = 15 \). Since \( t < 1 \), \( t = \frac{12}{y} < 1 \), and \( y > 12 \), then \( y = 15 \).

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