
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
Definition: If a number $p$ can be expressed as $p=x^{2}+y^{2}-xy$ (where $x$ and $y$ are positive integers), then $p$ is called a "Hilbert" number. For example, $39=7^{2}+5^{2}-7\times 5$, $147=13^{2}+11^{2}-13\times 11$, etc. Therefore, $39$ and $147$ are "Hilbert" numbers.
$(1)$ The rational number $1$ is ______ a "Hilbert" number (fill in "is" or "is not");
$(2)$ Numbers like $39$ and $147$ that can be expressed as the operation defined by two consecutive odd numbers are also called "$H$ Hilbert" numbers.
① Let the smaller of the two consecutive odd numbers be $2n-1$ (where $n$ is a positive integer), express the "$H$ Hilbert" number in terms of $n$ as ______;
② Given that the difference between two "$H$ Hilbert" numbers is $48$, then the larger of these two "$H$ Hilbert" numbers is ______.

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