
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
For the fraction "numerator is $1$, denominator can be written as the product of two positive factors," we can perform "splitting" transformation, for example: $\frac{1}{6}=\frac{1}{2×3}=\frac{1}{2}-\frac{1}{3}$; $\frac{1}{6}=\frac{1}{1×6}=\frac{1}{5}×(\frac{1}{1}-\frac{1}{6})$; $\frac{1}{18}=\frac{1}{3×6}=\frac{1}{3}×(\frac{1}{3}-\frac{1}{6})$; $\frac{1}{18}=\frac{1}{2×9}=\frac{1}{7}×(\frac{1}{2}-\frac{1}{9})$; $\ldots$ Using the above method, solve the following problems: (1) $\frac{1}{20}=\frac{1}{4×5}=(\frac{1}{4}-\frac{1}{5})$; $\frac{1}{20}=\frac{1}{2×10}=\_\_\_\_\_\_\times (\frac{1}{2}-\frac{1}{10})$; (2) If $\frac{1}{15}$ is transformed by splitting, then $\frac{1}{15}=$______; (3) Applying the above transformation, simplify: $\frac{1}{x(x+2)}+\frac{1}{(x+2)(x+4)}+\frac{1}{(x+4)(x+6)}+…+\frac{1}{(x+2022)(x+2024)}$.

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