writing very long function
Maybe it's my background of old C programmer but... I would just write
plain functions, no classes, like in C. And I would keep all this stuff
in a single file, to provide some form of encapsulation:

my_parser.h:

some_type get_numbers(int arg1, char arg2);


my_parser.cpp:

static int performFirstAction(float arg1, int arg2, char arg3)
{...}

static bool performSecondAction(long arg1, void *arg2, int arg3)
{...}

some_type get_numbers(int arg1, char arg2)
{...}


Note that there is only one function declared in the .h: this is the
“interface” to your module. Note also that all the other functions are
defined as static. This tells the compiler to keep them hidden from
the linking phase, so they do not pollute the global namespace. And
finally, note that there is no global shared variable: everything is
passed as arguments to the functions.

The rationale for the last point is that this helps the compiler
optimize things. A static function that is called only once is almost
guaranteed to be inlined by the compiler. Which means there is no
overhead in passing the arguments. On the other hand, if you use global
variables and the compiler doesn't manage to optimize them away, you
will incur the cost of RAM access.