
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
Observe the following equations:<br/>The $1$st equation: $a_{1}=\frac{1}{1+\sqrt{2}}=\sqrt{2}-1$,<br/>The $2$nd equation: $a_{2}=\frac{1}{\sqrt{2}+\sqrt{3}}=\sqrt{3}-\sqrt{2}$,<br/>The $3$rd equation: $a_{3}=\frac{1}{\sqrt{3}+2}=2-\sqrt{3}$,<br/>The $4$th equation: $a_{4}=\frac{1}{2+\sqrt{5}}=\sqrt{5}-2$,<br/>$\ldots $<br/>Following the above pattern, calculate $a_{1}+a_{2}+a_{3}+\ldots +a_{n}=\_\_\_\_\_\_$.

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