The optimization process and recommendations:

1. Avoid Correlated Subqueries (query line: 18): A correlated subquery is a subquery that contains a reference (column: t_col_2) to a table that also appears in the outer query. Usually correlated queries can be rewritten with a join clause, which is the best practice. The database optimizer handles joins much better than correlated subqueries. Therefore, rephrasing the query with a join will allow the optimizer to use the most efficient execution plan for the query.
2. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.

Optimal indexes for this query:

ALTER TABLE `actual_table` ADD INDEX `actual_table_idx_id_pid` (`ID`,`PID`);

The optimized query:

WITH temp_table (t_col_1, t_col_2, t_col_3) AS (SELECT
        actual_table.col_1 AS t_col_1,
        actual_table.col_2 AS t_col_2,
        actual_table.col_3 AS t_col_3 
    FROM
        actual_table 
    WHERE
        actual_table.ID = 100 
        AND actual_table.PID = 1245) SELECT
        t1.t_col_1,
        t1.t_col_2,
        t1.t_col_3 
    FROM
        temp_table AS t1 
    WHERE
        t1.t_col_2 BETWEEN 1 AND 12541 
        AND t1.t_col_1 = (
            SELECT
                max(t2.t_col_1) 
            FROM
                temp_table AS t2 
            WHERE
                t2.t_col_1 < 15147 
                AND t2.t_col_2 = t1.t_col_2
        ) 
    ORDER BY
        t1.t_col_2