SELECT 
  p.promotion_id, 
  p.from_date, 
  p.to_date, 
  active.from_date AS active_from_date, 
  active.to_date AS active_to_date, 
  expired.from_date AS expired_from_date, 
  expired.to_date AS expired_to_date 
FROM 
  cscart_promotions AS p 
  INNER JOIN cscart_ab__dotd AS ab__dotd ON p.promotion_id = ab__dotd.promotion_id 
  AND ab__dotd.use_schedule = 'Y' 
  LEFT JOIN (
    SELECT 
      promotion_id, 
      MIN(from_date) AS from_date, 
      MIN(to_date) AS to_date 
    FROM 
      cscart_ab__dotd_periods 
    WHERE 
      to_date >= 1754083484 
    GROUP BY 
      promotion_id
  ) AS active ON active.promotion_id = p.promotion_id 
  LEFT JOIN (
    SELECT 
      promotion_id, 
      MAX(from_date) AS from_date, 
      MAX(to_date) AS to_date 
    FROM 
      cscart_ab__dotd_periods 
    WHERE 
      to_date < 1754083484 
    GROUP BY 
      promotion_id
  ) AS expired ON expired.promotion_id = p.promotion_id 
WHERE 
  p.status IN ('A', 'H')

Query time 0.00053

JSON explain

{
  "query_block": {
    "select_id": 1,
    "cost_info": {
      "query_cost": "4.80"
    },
    "nested_loop": [
      {
        "table": {
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          ],
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          "rows_produced_per_join": 1,
          "filtered": "100.00",
          "cost_info": {
            "read_cost": "1.00",
            "eval_cost": "0.20",
            "prefix_cost": "1.20",
            "data_read_per_join": "40"
          },
          "used_columns": [
            "promotion_id",
            "use_schedule"
          ],
          "attached_condition": "(`44mm_ru`.`ab__dotd`.`use_schedule` = 'Y')"
        }
      },
      {
        "table": {
          "table_name": "p",
          "access_type": "eq_ref",
          "possible_keys": [
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          ],
          "key": "PRIMARY",
          "used_key_parts": [
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          ],
          "key_length": "3",
          "ref": [
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          ],
          "rows_examined_per_scan": 1,
          "rows_produced_per_join": 0,
          "filtered": "33.33",
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            "data_read_per_join": "26"
          },
          "used_columns": [
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            "to_date",
            "from_date",
            "status"
          ],
          "attached_condition": "(`44mm_ru`.`p`.`status` in ('A','H'))"
        }
      },
      {
        "table": {
          "table_name": "active",
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          "possible_keys": [
            "<auto_key0>"
          ],
          "key": "<auto_key0>",
          "used_key_parts": [
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          ],
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          "cost_info": {
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          },
          "used_columns": [
            "promotion_id",
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          ],
          "materialized_from_subquery": {
            "using_temporary_table": true,
            "dependent": false,
            "cacheable": true,
            "query_block": {
              "select_id": 2,
              "cost_info": {
                "query_cost": "1.20"
              },
              "grouping_operation": {
                "using_filesort": false,
                "table": {
                  "table_name": "cscart_ab__dotd_periods",
                  "access_type": "index",
                  "possible_keys": [
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                  ],
                  "key": "PRIMARY",
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                  ],
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                  ],
                  "attached_condition": "(`44mm_ru`.`cscart_ab__dotd_periods`.`to_date` >= 1754083484)"
                }
              }
            }
          }
        }
      },
      {
        "table": {
          "table_name": "expired",
          "access_type": "ref",
          "possible_keys": [
            "<auto_key0>"
          ],
          "key": "<auto_key0>",
          "used_key_parts": [
            "promotion_id"
          ],
          "key_length": "3",
          "ref": [
            "44mm_ru.ab__dotd.promotion_id"
          ],
          "rows_examined_per_scan": 2,
          "rows_produced_per_join": 1,
          "filtered": "100.00",
          "cost_info": {
            "read_cost": "1.33",
            "eval_cost": "0.27",
            "prefix_cost": "4.80",
            "data_read_per_join": "32"
          },
          "used_columns": [
            "promotion_id",
            "from_date",
            "to_date"
          ],
          "materialized_from_subquery": {
            "using_temporary_table": true,
            "dependent": false,
            "cacheable": true,
            "query_block": {
              "select_id": 3,
              "cost_info": {
                "query_cost": "1.20"
              },
              "grouping_operation": {
                "using_filesort": false,
                "table": {
                  "table_name": "cscart_ab__dotd_periods",
                  "access_type": "index",
                  "possible_keys": [
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                  ],
                  "key": "PRIMARY",
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                  "used_columns": [
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                    "to_date"
                  ],
                  "attached_condition": "(`44mm_ru`.`cscart_ab__dotd_periods`.`to_date` < 1754083484)"
                }
              }
            }
          }
        }
      }
    ]
  }
}