
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
Let \( H_{n} = \{ \sqrt{2 \pm \sqrt{2 \cdots \pm \sqrt{2}}} \} \), where the number of square roots is \( n \). How many real elements does the set \( \bigcup_{n=1}^{2000} H_{n} \) have?

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