
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 \(a > b\), where \(x, y, z\) are digits. Consider the following cases for the decimal representation of the desired numbers.

Case 1. Let \(a = x \cdot 10 + y\), \(b = y \cdot 10 + z\).

a) Write the conditions for these notations:
\[
\begin{cases}
10x + y = 3(10y + z), \\
x + y = y + z + 3,
\end{cases} 
\quad \begin{cases}
10x - 29y = 3z, \\
z = x - 3,
\end{cases} \quad 7x - 29y = -9,
\quad \begin{cases}
x = 7 \\
y = 2 \\
z = 4
\end{cases}
\]
Thus, \(a = 72\), \(b = 24\).

b)
\[
\begin{cases}
10x + y = 3(10y + z), \\
x + y = y + z - 3,
\end{cases} \quad \begin{cases}
10x - 29y = 3z, \\
z = x + 3,
\end{cases} \quad 7x - 29y = 9, \text{ no digits satisfy the conditions}.
\]

Case 2. Let \(a = x \cdot 10 + y\), \(b = z \cdot 10 + x\).
\[
\begin{cases}
10x + y = 3(10z + x), \\
x + y = x + z \pm 3,
\end{cases} \quad \begin{cases}
7x + y = 30z, \\
z = y \pm 3,
\end{cases} \quad 7x - 29y = \pm 90, \emptyset
\]

Case 3. Let \(a = x \cdot 10 + y\), \(b = x \cdot 10 + z\), no digits satisfy the conditions.

Case 4. Let \(a = x \cdot 10 + y\), \(b = z \cdot 10 + y\).
\[
\begin{aligned}
\begin{cases}
10x + y = 3(10z + y), \\
x + y = z + y \pm 3,
\end{cases} \quad \begin{cases}
10x - 2y = 30z, \\
z = x \pm 3,
\end{cases} \quad 10x + y = \pm 45, \\
\begin{cases}
x = 4 \\
y = 5 \\
z = 1
\end{cases}
\end{aligned}
\]
Thus, \(a = 45\), \(b = 15\).

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