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I notice the SoftwareSerial library being used, but software serial is not used in the program on pins 7,8 so the code is not communicating with the shield, only with the serial monitor. "it wont work this way. You need to add:SoftwareSerial mySerial(7, 8);
That code wont work, because the use of Software Serial was not implemented. code found here will work. //https://www.elecrow.com/download/Using_SMS_to_Control_Relay_Hardware_Serial.ino
I had the similar problems. one big one is nothing in the setup() function would print. Then i found that you 'need' to use while(!Serial); right after the Serial.begin(...); Then everything worked fine, and the serial port issues became very bearable. i found it in a blog, and i am so thankful, forgot his name...Hope this helps_
it clearly says to use the modem port per mediatek "getting started" tutorial.
The code is missing the three #include header files "it does not compile.