
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
Read the following material:<br/>Let the areas of squares with side lengths $a$, $a+\sqrt{b}$, $a+2\sqrt{b}$, $a+3\sqrt{b}$ be denoted as $S_{1}$, $S_{2}$, $S_{3}$, $S_{4}$, respectively.<br/>Then $S_{2}-S_{1}=(a+\sqrt{b})^{2}-a^{2}$<br/>$=[(a+\sqrt{b})+a]\cdot [(a+\sqrt{b})-a]$<br/>$=(2a+\sqrt{b})\cdot \sqrt{b}$<br/>$=b+2a\sqrt{b}$<br/>For example, when $a=1$, $b=3$, $S_{2}-S_{1}=3+2\sqrt{3}$<br/>Based on the above material, answer the following questions:<br/>$(1)$ When $a=1$, $b=3$, $S_{3}-S_{2}=$______, $S_{4}-S_{3}=$______; <br/>$(2)$ When $a=1$, $b=3$, the area of a square with side length $a+n\sqrt{b}$ is denoted as $S_{n+1}$, where $n$ is a positive integer. From the calculations in (1), can you guess what $S_{n+1}-S_{n}$ is equal to? Provide a proof for your guess; <br/>$(3)$ When $a=1$, $b=3$, let $t_{1}=S_{2}-S_{1}$, $t_{2}=S_{3}-S_{2}$, $t_{3}=S_{4}-S_{3}$, ..., $t_{n}=S_{n+1}-S_{n}$, and $T=t_{1}+t_{2}+t_{3}+\ldots +t_{50}$, find the value of $T$.

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