
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
There is a set of numbers with a pattern: $1$, $-1$, $\sqrt{2}$, $-\sqrt{2}$, $\sqrt{3}$, $-\sqrt{3} \ldots \ldots $, where $1$, $-1$, $\sqrt{2}$, $-\sqrt{2}$, $\sqrt{3}$, $-\sqrt{3}$ repeat according to this pattern.<br/>$(1)$ The $10$th number is ______, and the $50$th number is ______.<br/>$(2)$ Starting from the first number, if the squares of consecutive numbers are added up to $520$, how many numbers' squares are added up in total?

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