
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 smallest solution to the inequality

$$
\frac{-\log _{2}(105+2 x \sqrt{x+19})^{3}+\left|\log _{2} \frac{105+2 x \sqrt{x+19}}{\left(x^{2}+x+3\right)^{4}}\right|}{9 \log _{5}(76+2 x \sqrt{x+19})-4 \log _{2}(105+2 x \sqrt{x+19})} \geqslant 0
$$

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