
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
Given the system of equations:
1. \( x + 2 + \sqrt{x^2 + 4x + y - 2} = 0 \)
2. \(\log_{|x+4|}(ax+5a) = 2\log_{|x-4|}(x+y)\)

Find the values of the parameter \(a\) such that the equation \( x^2 - (a - 12)x + 36 - 5a = 0 \) has two distinct solutions, satisfying:

- \( -6 < x \leq -2 \)
- \( x \neq -5 \)
- \( x \neq -4 \)
- \( x \neq -3 \)

Also, determine for which values of \(a\) the points \( x = -5, x = -4, x = -3 \) are solutions to the given equation.

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