Generate program that has the following intention:

This is the structure of the `NAME_1` class:
````
class NAME_1:
    def __init__(
        self,
        value=None,
        successor=None,
        successors=[],
        predecessors=[],
        incoming_nodes=[],
        outgoing_nodes=[],
    ):
        self.value = value
        self.successor = successor
        self.successors = successors
        self.predecessors = predecessors
        self.incoming_nodes = incoming_nodes
        self.outgoing_nodes = outgoing_nodes
````

The intention of the program is to implement a depth-first search (DFS) algorithm for solving a problem, represented by the NAME_1 class. The NAME_1 class has a directed graph structure with successors, predecessors, and incoming/outgoing nodes. The p4 function is a depth-first search function that takes a start NAME_1 and a goal NAME_1, and searches for a path from the start NAME_1 to the goal NAME_1. The search is performed using the search_from function, which is a recursive depth-first search function.
 Use words in all capital letters at least 4 times to highlight key points.
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