
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 to the equation under the domain restrictions \( x \neq 2 \), \( x \neq 6 \), and \( a - 7x + 39 \geq 0 \):

\[ x^{2}-4x-21+\left(\frac{|x-2|}{x-2}+\frac{|x-6|}{x-6}+a\right)^{2}=0 \]

or

\[ (x-2)^{2}+\left(\frac{|x-2|}{x-2}+\frac{|x-6|}{x-6}+a\right)^{2}=25. \]

Consider different cases based on the value of \( x \) and find the suitable values of \( x \) parameterized by \( a \):

1. \( x < 2 \), solve for \( x \) in:

\[ (x-2)^{2}+(a-2)^{2}=25. \]

2. \( 2 < x < 6 \), solve for \( x \) in:

\[ (x-2)^{2}+a^{2}=25. \]

3. \( x > 6 \), solve for \( x \) in:

\[ (x-2)^{2}+(a+2)^{2}=25. \]

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