
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
If a series of functions have the same analytic expression and range but different domains, they are referred to as "twin functions." Therefore, how many twin functions are there whose expression is $y=2x^2+1$ and range is $\{5,19\}$?

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