
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

Solve the inequality \(\frac{\sqrt{x^{2}+x}-\sqrt{4-2 x}}{2 x+5-2 \sqrt{x^{2}+5 x+6}} \leq 0\). 

Consider the domain of validity:

- \(x(x+1) \geq 0\)
- \(4-2 x \geq 0\)
- \(x^{2}+5 x+6 \geq 0\)

Which gives \(x \in (-\infty, -3] \cup [-2, -1] \cup [0, 2]\).

The given inequality is equivalent to:

\[
\frac{(x+4)(x-1)}{(x+3)-2 \sqrt{(x+3)(x+2)}+x+2} \leq 0.
\]

For \(x \geq -2\):

\[
\frac{(x+4)(x-1)}{(\sqrt{x+3}-\sqrt{x+2})^{2}} \leq 0,
\]

which gives \(x \in [-2, 1]\). Considering the domain of validity, we get \(x \in [-2, -1] \cup [0, 1]\).

For \(x \leq -3\):

\[
\frac{(x+4)(x-1)}{-(\sqrt{-x-3}+\sqrt{-x-2})^{2}} \leq 0,
\]

which gives \(x \in (-\infty, -4]\), which is included in the domain. 

Thus, the final solution is \(x \in (-\infty, -4] \cup [-2, -1] \cup [0, 1]\).

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