
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 function calculator with a display screen and two operation buttons. If you press the first button, the number on the display becomes $\left[\frac{x}{2}\right]$ (where $[x]$ denotes the greatest integer less than or equal to the real number $x$). If you press the second button, the number on the display becomes $4x+1$. Pressing either button once is considered one operation. Currently, the number on the display is 1. 

(1) Is it possible for the number 2000 to appear on the display after a finite number of operations? Provide a justification.

(2) How many integers less than 2000 can appear on the display after a finite number of operations?

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