
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
Explore the pattern, observe the following equations, and answer the questions.  
$1+3=4=2^{2}$; $1+3+5=9=3^{2}$; $1+3+5+7=16=4^{2}$; $1+3+5+7+9=25=5^{2}$.  
$(1)$ Please guess: $1+3+5+7+9+\ldots +99=$______;  
$(2)$ Please guess: $1+3+5+7+9+\ldots +(2n-1)=\_\_\_\_\_\_;(n$ is a positive integer and $n\geqslant 1)$  
$(3)$ Calculate: $201+203+205+\ldots +395+397+399$.

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