
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
The quadratic equation \( a^{2} - 18a + 72 = 0 \) has solutions \( a = 6 \) and \( a = 12 \). Then, for \( x^{2} - 12x = 0 \), \( x_{1} = 0 \) (an extraneous root) and \( x_{2} = 12 \) satisfy the conditions. Similarly, for \( x^{2} - 24x = 0 \), \( x_{1} = 0 \) (an extraneous root) and \( x_{2} = 24 \) satisfy the conditions.

Consider the cases when the system will have two distinct solutions. The quadratic equation \((*)\) will have two distinct positive roots \( x_{1,2} = a \pm \sqrt{18(a-4)} \) if:

\[
\left\{
\begin{array}{c}
18(a - 4) > 0, \\
a > 0, \\
a^2 - 18a + 72 > 0 
\end{array}
\right.
\]

\[
\Leftrightarrow 
\left\{
\begin{array}{c}
a > 4, \\
\left[
\begin{array}{c}
a < 6, \\
a > 12
\end{array}
\right
\end{array}
\right.
\Leftrightarrow 
\left\{
\begin{array}{c}
4 < a < 6, \\
a > 12
\end{array}
\right.
\]

From this set, we remove the previously considered point \( a = 22 \).

Combining the found values of \( a \), we get the answer.

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