
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
Consider the cases where the system will have two distinct solutions. The quadratic equation (*) will have two distinct positive roots \(X_{1,2} = a \pm 3 \sqrt{a-2}\), if

$$
\left\{
\begin{array} { c } 
{ 9 ( a - 2 ) > 0 , } \\
{ \quad a > 0 , } \\
{ a ^ { 2 } - 9 a + 18 > 0 }
\end{array} 
\Leftrightarrow 
\left\{
\begin{array} { l } 
{ a > 2 } \\
{ [ \begin{array} { l } 
{ a < 3 , } \\
{ a > 6 }
\end{array} 
\end{array} 
\Leftrightarrow 
\left\{
\begin{array}{c}
2<a<3 \\
a>6
\end{array}
\right.
\right.
\right.
$$

From this set, we need to exclude the point \(a=11\).

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

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