Why are my If else statements not working?

This program asks for a number from 1 to 12 and prints out the month and season correspondent to the input number, if the user types anything else than numbers from 1 to 12 an error message is displayed. It looks like the first if statement runs perfectly with its correspondent nested statements but the next else if statements wont work, try with 1, 2, and 12 and the program will work perfectly, but it doesn't with any other value, does anyone knows why this is happening? Here's the program...

#include <iostream>
#include <string>
using namespace  std;

int main() {

  int number;
  string season, month;

  cout << "Welcome! This program will provide the season of the year based on the month you enter, 1 corresponding to January, 2 for February and so on until 12 for December" << endl << endl;

  cout << "Enter a number between 1 and 12: ";
  cin >> number;

  if (number == 1 || 2 || 12)
  {
    if (number == 1)
    {
      month = "January";
      season = "winter";
    }
    else if (number == 2)
    {
      month = "February";
      season = "winter";
    }
    else if (number == 12)
    {
      month = "December";
      season = "winter";
    }
  }
  else if (number == 3 || 4 || 5)
  {
    if (number == 3)
    {
      month = "March";
      season = "spring";
    }
    else if (number == 4)
    {
      month = "April";
      season = "spring";
    }
    else if (number == 5)
    {
      month = "May";
      season = "spring";
    }
  }
  else if (number == 6 || 7 || 8 )
  {
    if (number == 6)
    {
      month = "June";
      season = "summer";
    }
    else if (number == 7)
    {
      month = "July";
      season = "summer";
    }
    else if (number == 8)
    {
      month = "August";
      season = "summer";
    }
  }
  else if (number == 9 || 10 || 11)
  {
    if (number == 9)
    {
      month = "September";
      season = "fall";
    }
    else if (number == 10)
    {
      month = "October";
      season = "fall";
    }
    else if (number == 11)
    {
      month = "November";
      season = "fall";
    }
  }
  else
  {
    cout << "You entered an ivalid value";
    return 0;
  }

  cout << "-------------------------------------------------------------" << endl;
  cout << "You entered " << number << ". " << month << " is the " << season << " season.";
  return 0;
}