
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

Transform the expression:

\[ \frac{x^{3}+4x^{2}-x-4}{(x+2)(x+4)-3x-12}=\frac{x^{2}(x+4)-(x+4)}{(x+2)(x+4)-3(x+4)}=\frac{(x+4)(x-1)(x+1)}{(x+4)(x-1)}=x+1 \]
\[ f(x)=\left|\frac{x^{3}+4x^{2}-x-4}{(x+2)(x+4)-3x-12}\right|=|x+1| \]
\[ D(f): x \neq -4, x \neq 1 \]

Given:

\[ p(x)=\sqrt{x^{2}-8x+16}+a=|x-4|+a \]

we obtain the equation:

\[ |x+1|=|x-4|+a \]

Solve graphically in the XOA coordinate system:

\[ 
\begin{cases}
x<-1 \\
-x-1=-x+4+a
\end{cases}
\Rightarrow
\begin{cases}
x<-1 \\
a=-5
\end{cases}
\]

\[ 
\begin{cases}
-1 \leq x<4 \\
x+1=-x+4+a
\end{cases}
\Rightarrow
\begin{cases}
1 \leq x<4 \\
a=2x-3
\end{cases}
\]

The equation has 1 solution if:

\[ a \in (-5, -1) \cup (-1, 5) \]

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