
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:

$$
\frac{x^{3}+4 x^{2}-x-4}{(x+2)(x+4)-3 x-12}=\frac{x^{2} (x+4)-(x+4)}{(x+2)(x+4)-3(x+4)}=\frac{(x+4)(x-1)(x+1)}{(x+4)(x-1)}=x+1
$$

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

and 

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

we obtain the equation: \( |x+1| = |x-4| + a \).

Solve it graphically:

For \( x < -1 \):
\[
-x-1 = -x+4+a \Rightarrow a = -5
\]

For \( -1 \leq x < 4 \):
\[
x+1 = -x+4+a \Rightarrow a = 2x - 3
\]

For \( x \geq 4 \):
\[
x+1 = x+4+a \Rightarrow a = 5
\]

The equation has one solution when \( a \in (-5, -1) \cup (-1, 5) \).

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