
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 inequality

\[ \frac{(\sqrt{x+6}-1)(\sqrt{x+6}-3)\left(\log _{2}(x+1)-\log _{2} 1\right)}{\left(2^{x}-1\right)\left(2^{x}-2\right)\left(\log _{5}(5-x)-\log _{5} 1\right)} \geq 0, \]

find the solution for \( x \), considering the domain restrictions \( x+6 \geq 0 \), \( x+1>0 \), \( 5-x>0 \), \( x \neq 0 \), \( x \neq 1 \), \( x \neq 4 \).

The domain is \( x \in(-1 ; 0) \cup(0 ; 1) \cup(1 ; 4) \cup(4 ; 5) \).

After considering these restrictions, the equivalent inequality to solve is:

\[ \frac{(x+5)(x-3) x}{x(x-1)(4-x)} \geq 0 \]

or simplified,

\[ \frac{(x-3)}{(x-1)(4-x)} \geq 0 \]

Find the solution set for \( x \).

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