
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]


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Find the value of:<br/>$(1)$ If real numbers $x$ and $y$ satisfy $\left(2x+y\right)^{2}-8\left(2x+y\right)-9=0$, then the value of $2x+y$ is ______;<br/>$(2)$ If real numbers $x$ and $y$ satisfy $2x+y-9=8\sqrt{2x+y}$, find the value of the algebraic expression $2x+y$;<br/>$(3)$ If real number $x$ satisfies $(x^{2}+2x)^{2}+4(x^{2}+2x)-5=0$, find the value of the algebraic expression $x^{3}+3x^{2}+x$.

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