
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
In a math class, Teacher Zhang wrote down the following four equations. By carefully observing the following equations, what pattern can you find: $1\times 3+1=2^{2}$, $2\times 4+1=3^{2}$, $3\times 5+1=4^{2}$, $4\times 6+1=5^{2}$, $\ldots$<br/>$(1)$ Please write down two more equations according to this pattern: ______, ______.<br/>$(2)$ Please express the pattern you found using an equation containing only the letter $n$ ($n$ is a positive integer): The pattern you found is ______.<br/>$(3)$ Please explain the correctness of this equation using the knowledge you have learned: ______.

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