Implement the random Fibonacci sequence The random Fibonacci sequence is defined as follows:
$$
f_n = 
\begin{cases}
f_{n-1}+f_{n-2} \text{ with probability } 1/2 \\
f_{n-1}-f_{n-2} \text{ with probability } 1/2 \\
\end{cases}
$$
$$
f_1 = f_2 = 1
$$
i.e. whether the next term is the sum or difference of the previous two is chosen at random, independently of previous terms. Your task is to implement this sequence.
Each random realization of the sequence must use consistent values. For example, if \$f_3 = 2\$, \$f_4\$ must then be either \$2+1 = 3\$ or \$2-1 = 1\$. This can be thought of as the sequence &quot;remembering&quot; previous values. This means that this example program is invalid, as previous values in the sequence are not maintained by later values. Furthermore, you should explain how your program meets the \$1/2\$ probability requirement.
As is standard for sequence challenges, you can perform one of three tasks:

Take a positive integer \$n\$ as input and output \$f_n\$
Take a positive integer \$n\$ as input and output \$f_1, f_2, ..., f_n\$
Output the sequence indefinitely with no end

Again, as is standard, you may use either \$0\$ or \$1\$ indexing, but the two initial values \$f_1 = f_2 = 1\$ must be used.
This is code-golf, so the shortest code, in bytes, wins.
Examples
n -&gt; possible values of f_n | probabilities of values
1 -&gt; 1                      | 1
2 -&gt; 1                      | 1
3 -&gt; 2, 0                   | 1/2, 1/2
4 -&gt; 3, 1, -1               | 1/4, 1/2, 1/4
5 -&gt; 5, 3, 1, -1            | 1/8, 1/8, 3/8, 3/8
6 -&gt; 8, 4, 2, 0, -2         | 1/16, 1/8, 1/4, 5/16, 1/4
05AB1E, 8 7 bytes
λ₂D(‚Ω+

-1 byte thanks to @ovs.
Prints the infinite sequence.
Try it online.
Explanation:

λ        # Create a recursive environment to output the infinite sequence,
         # implicitly starting at a(0)=1
         #  (push a(n-1) implicitly)
 ₂       #  Push a(n-2) (NOTE: all negative a(n) are 0, so a(-1)=0)
  D      #  Duplicate a(n-2)
   (     #  Negate the copy: -a(n-2)
    ‚    #  Pair them together: [a(n-2), -a(n-2)]
     Ω   #  Pop and push a random item
      +  #  And add it to the a(n-1)
         # (after which the infinite list is output implicitly)