
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
Let $\left\{x_{n}\right\}$ be a sequence of positive real numbers. If $x_{1}=\frac{3}{2}$ and $x_{n+1}^{2}-x_{n}^{2}=\frac{1}{(n+2)^{2}}-\frac{1}{n^{2}}$ for all positive integers $n$, find $x_{1}+x_{2}+\cdots+x_{2009}$.

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