
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 when the system will have two distinct solutions. The quadratic equation will have two distinct positive roots \( x_{1,2} = a \pm 4 \sqrt{a-3} \) if

\[
\left\{
\begin{array}{c}
16(a-3) > 0 \\
a > 0 \\
a^2 - 16a + 48 > 0
\end{array}
\right.
\Leftrightarrow 
\left\{
\begin{array}{c}
a > 3 \\
a < 4 \\
a > 12
\end{array}
\right.
\Leftrightarrow 
\left\{
\begin{array}{c}
3 < a < 4 \\
a > 12
\end{array}
\right.
\]

From this set, we need to remove the previously considered point \( a = 19 \).

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

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