
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
Find the solution for the equation: $x \neq 1, x \neq 5$, $a-7x+32 \geq 0$. 
1. For $a-7x+32=0$, solve for $x: x = \frac{a+32}{7}$.
2. Solve $x^{2}-2x-24+\left(\frac{|x-1|}{x-1}+\frac{|x-5|}{x-5}+a\right)^{2}=0$ or $(x-1)^{2}+\left(\frac{|x-1|}{x-1}+\frac{|x-5|}{x-5}+a\right)^{2}=25$.

Expand the absolute values:
- For $x<1$: $(x-1)^{2}+(a-2)^{2}=25$, $x=1-\sqrt{25-(a-2)^{2}}$.
- For $1<x<5$: $(x-1)^{2}+a^{2}=25$, $x=1+\sqrt{25-a^{2}}$.
- For $x>5$: $(x-1)^{2}+(a+2)^{2}=25$, $x=1+\sqrt{25-(a+2)^{2}}$. 

Graph the resulting functions in the $xOy$ coordinate system, taking into account the constraint $a-7x+32 \geq 0$ and excluding points where $x=1$ and $x=5$. Parallel lines to the $Ox$ axis intersect the marked curves exactly at two points for $a \in(-5, -4) \cup(-3, 3) \cup[5, 7)$.

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