3. A function calculator has a display screen and two operation keys. If the first operation key is pressed, the number $x$ on the display will change to $\left[\frac{x}{2}\right]$ (where $[x]$ represents the greatest integer not exceeding $x$); if the second operation key is pressed, the number $x$ on the display will change to $4x+1$. Pressing any operation key is considered one operation. Now the number on the display is 1. The questions are:
(1) Can the number 2000 appear on the display after a finite number of operations? Explain your reasoning;
(2) How many integers less than 2000 can appear on the display after a finite number of operations?
(1) When the number $x$ is represented in binary, pressing the first operation key is equivalent to removing the last digit of the binary representation of $x$; pressing the second operation key is equivalent to adding 01 at the end of the binary representation of $x$. Since the initial value is 1, the digit 1 in the binary representation of the resulting number cannot be adjacent. Since $2000=2^{10}+2^{9}+2^{8}+2^{7}+2^{6}+2^{4}=11111010000_{(2)}$, it is impossible to display the integer 2000 on the screen after a finite number of operations.
(2) From the analysis in (1), any number $x$ whose binary representation has no adjacent 1s can be obtained through a finite number of operations, and the largest number less than 2000 is $10101010101_{(2)}=1365$. Among these, there is 1 number with no 1s; there are $C_{11}^{1}=11$ numbers with 1 one; there are $C_{10}^{2}=45$ numbers with 2 ones; there are $C_{9}^{3}=84$ numbers with 3 ones; there are $C_{8}^{4}=70$ numbers with 4 ones; there are $C_{7}^{5}=21$ numbers with 5 ones; there is $C_{6}^{6}=1$ number with 6 ones.
Therefore, there are a total of 233 numbers that can appear on the screen after a finite number of operations.