It is possible to call PHP functions from SQLite. It makes it possible to use simple queries to 'normalize' the table.
I have converted the 'normalize  comma delimited strings holding keys' to SQLite. see: (joining on ';' separated values in a column) for a more complete explanation
I started out looking for ways of converting the functions to run in SQLite. After some searching I came across this: Working with PHP UDFs in SQLite.
Which I found interesting - call PHP functions from SQLite. That sounds like fun!
It works but you cannot use PDO. You have to use the SQLite functions directly.  Vendor Specific Database Extensions: SQLite3
Updated with your data (see previous edits for working code for other question)
The code: 
<?php // Q34231542 -- count_in_set, value_in_set
/*
 * See this question for a rather more complete explanation of what this is doing...
 *
 * https://stackoverflow.com/questions/33782728/can-i-resolve-this-with-pure-mysql-joining-on-separated-values-in-a-column/
 */
define('SQLITE_DB', __DIR__ .'/Q34231542.sqlite');
/**
 * @var SQLite3
 */
$db = new SQLite3(SQLITE_DB);
/*
 * Define the functions for use by SQLite.
 */
$db->createFunction('count_in_set', 'count_in_set', 2);
$db->createFunction('value_in_set', 'value_in_set', 3);
$sql ="
SELECT STRATEGY.STRATEGY_ID                         as 'applied_strategy_id',
       STRATEGY.APPLIED_SET_IDS                     as 'applied_strategy_list',
       isequence.id                                 as 'which_strategy',
       COUNT_IN_SET(STRATEGY.APPLIED_SET_IDS, ',')  as 'StrategyCount',
       VALUE_IN_SET(STRATEGY.APPLIED_SET_IDS, ',', isequence.id)
                                                    as 'TheStrategy',
       ACTION_SET.VALUE                             as 'Action_Set_Value'
FROM STRATEGY
     JOIN integerseries AS isequence
       ON isequence.id <= COUNT_IN_SET(STRATEGY.APPLIED_SET_IDS, ',') /* normalize */
     JOIN ACTION_SET
       ON ACTION_SET.APPLIED_ACTION_SET_ID = VALUE_IN_SET(STRATEGY.APPLIED_SET_IDS, ',', isequence.id)
ORDER BY
       STRATEGY.STRATEGY_ID , ACTION_SET.APPLIED_ACTION_SET_ID;
";
/*
 * Run the query
 */
$stmt   = $db->prepare($sql);
$result = $stmt->execute();
/*
 * Get the results
 */
$rows = array();
while ($row = $result->fetchArray(SQLITE3_ASSOC)) { // fetch all the rows for now...
    $rows[] = $row;
}
/*
 * output...
 */
 // \Kint::dump($rows);
 echo '<pre>';
 var_dump($rows);
 echo '</pre>';
exit;
/* -------------------------------------------------------------------------
 *  The PHP functions called from SQLite
 */
/**
 * Count the number of delimited items in a string
 *
 * @param string $delimitedValues
 * @param string $delim
 * @return integer
 */
function count_in_set($delimitedValues, $delim)
{
    return substr_count(trim($delimitedValues, $delim), $delim) + 1;
}
/**
 * Treat the delimited values as ONE BASED array.
 *
 * @param string $delimitedValues
 * @param string $delim
 * @param integer $which
 * @return string
 */
function value_in_set($delimitedValues, $delim, $which)
{
    $items = explode($delim, $delimitedValues);
    return $items[$which - 1];
}
The output:
    applied_strategy_id applied_strategy_list which_strategy StrategyCount TheStrategy Action_Set_Value
#1  1                   "1,3,6,7"             1              4            "1"         "X"
#2  1                   "1,3,6,7"             2              4            "3"         "Z"
#3  1                   "1,3,6,7"             3              4            "6"         "J"
#4  1                   "1,3,6,7"             4              4            "7"         "K"
#5  2                   "1,2,4"               1              3            "1"         "X"
#6  2                   "1,2,4"               2              3            "2"         "Y"
#7  2                   "1,2,4"               3              3            "4"         "H"
The data:
CREATE TABLE [integerseries] (
  [id] INTEGER NOT NULL PRIMARY KEY);
INSERT INTO "integerseries" VALUES(1);
INSERT INTO "integerseries" VALUES(2);
INSERT INTO "integerseries" VALUES(3);
INSERT INTO "integerseries" VALUES(4);
INSERT INTO "integerseries" VALUES(5);
INSERT INTO "integerseries" VALUES(6);
INSERT INTO "integerseries" VALUES(7);
INSERT INTO "integerseries" VALUES(8);
INSERT INTO "integerseries" VALUES(9);
INSERT INTO "integerseries" VALUES(10);
CREATE TABLE STRATEGY (STRATEGY_ID INTEGER PRIMARY KEY, APPLIED_SET_IDS VARCHAR);
INSERT INTO "STRATEGY" VALUES(1,'1,3,6,7');
INSERT INTO "STRATEGY" VALUES(2,'1,2,4');
CREATE TABLE ACTION_SET (APPLIED_ACTION_SET_ID INTEGER PRIMARY KEY, VALUE VARCHAR);
INSERT INTO "ACTION_SET" VALUES(1,'X');
INSERT INTO "ACTION_SET" VALUES(2,'Y');
INSERT INTO "ACTION_SET" VALUES(3,'Z');
INSERT INTO "ACTION_SET" VALUES(4,'H');
INSERT INTO "ACTION_SET" VALUES(5,'I');
INSERT INTO "ACTION_SET" VALUES(6,'J');
INSERT INTO "ACTION_SET" VALUES(7,'K');
INSERT INTO "ACTION_SET" VALUES(8,'L');