Solution:
On R1:
R1(config)#router bgp 123
R1(config-router)#address-family ipv4
R1(config-router-af)#neighbor 10.0.0.2 allowas-in
On R3:
R3(config)#router bgp 123
R3(config-router)# address-family ipv4
R3(config-router-af)#neighbor 192.168.1.2 allowas-in
VERIFICATION:
R3#sh ip route bgp
Gateway of last resort is not set
1.0.0.0/32 is subnetted, 1 subnets
B 1.1.1.1 [20/0] via 192.168.1.2, 00:01:17
2.0.0.0/32 is subnetted, 1 subnets
B 2.2.2.2 [20/0] via 192.168.1.2, 00:05:06
10.0.0.0/24 is subnetted, 1 subnets
B 10.0.0.0 [20/0] via 192.168.1.2, 00:01:17
Test Ping from R3 to R1:
R3#ping 1.1.1.1
Type escape sequence to abort.
Sending 5, 100-byte ICMP Echos to 1.1.1.1, timeout is 2 seconds:
!!!!!
R3#ping 1.1.1.1 source lo0
Type escape sequence to abort.
Sending 5, 100-byte ICMP Echos to 1.1.1.1, timeout is 2 seconds:
Packet sent with a source address of 3.3.3.3
!!!!!
Success rate is 100 percent (5/5), round-trip min/avg/max = 1/1/1 ms

Explanation
According to the requirements (first use TACACS+, then allow login with no authentication), we have to use "aaa authentication login ... group tacacs+ none" for AAA command.
The next thing to check is the if the "aaa authentication login default" or "aaa authentication login list-name" is used. The 'default' keyword means we want to apply for all login connections (such as tty, vty, console and aux). If we use this keyword, we don't need to configure anything else under tty, vty and aux lines. If we don't use this keyword then we have to specify which line(s) we want to apply the authentication feature.
From above information, we can find out answer 'R1#sh run | include aaa aaa new-model aaa authentication login default group tacacs+ none aaa session-id common R1#sh run | section vty line vty 0 4 password 7 0202039485748 If you want to learn more about AAA configuration, please read our AAA TACACS+ and RADIUS Tutorial - Part 2.
For your information, answer 'R1#sh run | include aaa
aaa new-model
aaa authentication login telnet group tacacs+ none
aaa session-id common
R1#sh run | section vty
line vty 0 4
R1#sh run | include username
R1#' would be correct if we add the following command under vty line ("line vty 0 4"): "login authentication telnet" ("telnet" is the name of the AAA list above)