Path graph $U$ is given, and a blindfolded player is standing on one of its vertices. The vertices of $U$ are labeled with positive integers between 1 and $n$, not necessarily in the natural order. In each step of the game, the game master tells the player whether he is in a vertex with degree 1 or with degree 2. If he is in a vertex with degree 1, he has to move to its only neighbour, if he is in a vertex with degree 2, he can decide whether he wants to move to the adjacent vertex with the lower or with the higher number. All the information the player has during the game after $k$ steps are the $k$ degrees of the vertices he visited and his choice for each step. Is there a strategy for the player with which he can decide in finitely many steps how many vertices the path has?
To solve the problem, we need to determine the number of vertices in a path graph \( U \) by following a strategy that allows a blindfolded player to gather enough information through a series of moves. The vertices of \( U \) are labeled with positive integers between 1 and \( n \), not necessarily in the natural order. The player is informed about the degree of the vertex they are currently on and can make decisions based on this information.

### Part 1: Algorithm \( A_0 \) to Reach an Endpoint

1. **Base Case \( A_3 \)**:
   - If the player is at a vertex with degree 1, they are at an endpoint.
   - If the player is at a vertex with degree 2, they should move to the adjacent vertex with the lower number.

2. **Inductive Step \( A_{i+1} \) from \( A_i \)**:
   - Assume the player is on a path with \( i+1 \) vertices at a point \( 1 < k < i+1 \).
   - Run algorithm \( A_i \). If the player reaches \( P_{i+1} \), they are done.
   - If the player reaches \( P_i \), they "bump" back to \( P_{i-1} \), acting as if they are at a degree 1 vertex.
   - Repeat the process until the player either finishes or ends up at \( P_2 \) or \( P_i \).
   - If the player is at \( P_2 \), run \( A_i \) again. If they do not finish, they will be at \( P_2 \) or \( P_i \).
   - If the player is still at \( P_2 \) or \( P_i \) after multiple runs, they can use the complement of the first step of \( A_i \) to reach \( P_1 \).

3. **Combining to Form \( A_0 \)**:
   - Combine the algorithms \( A_3, A_4, A_5, \ldots \). If there are \( N \) vertices, the algorithm will finish before \( A_{N+1} \) is started.

### Part 2: Determining the Number of Vertices

1. **Apply Algorithm \( A_0 \)**:
   - Use \( A_0 \) to reach an endpoint, \( P_1 \).

2. **Building the Diagram**:
   - Move from \( P_1 \) to \( P_2 \) and determine whether \( P_{i-1} \) or \( P_{i+1} \) is larger for all \( 2 \leq i \leq n-1 \).

3. **Case Analysis for \( i=2 \)**:
   - Move to the lower number from \( P_2 \).
   - If the lower number leads to a vertex of degree 1, either \( n=3 \) or \( P_1 < P_3 \). Verify by moving to \( P_2 \) and then to the higher number.
   - If the lower number leads to a vertex of degree 2, then \( P_1 > P_3 \) and \( n > 3 \).

4. **Building Diagram for \( i=k+1 \) from \( 2 \leq i \leq k \)**:
   - After knowing the diagram for \( i=k \), apply \( A_0 \) again to reach an endpoint.
   - Test whether the endpoint has the same diagram by moving to \( Q_2 \) and checking expectations.
   - If the diagram is the same, continue building up the diagram.
   - If the diagram is different, gather partial information about both endpoints and continue building up the diagrams.

5. **Finishing**:
   - Continue building the diagrams until the player finds that \( S_n \) has degree 1, indicating the path has \( n \) vertices.

The final answer is \( \boxed{ n } \).