Given a graph $ G(V, E)$ with capacities on the edges such that all edges have a capacity of $ sqrt(2)$ apart from one edge with a capacity of 2. need to find max flow efficiently.

I can run Dinic on this graph or FordFulkerson but I know it can be more time-efficient.

What I have tried –

transform all edges capacity to be 1, then find max flow using Dinic algorithm on 0-1 network that is more efficient than a general network. then If the edge that had a capacity of 2 isn’t saturated it won’t be saturated on the original graph so we can just find the min-cut and multiply the number of edges that cross the min-cut by sqrt(2) and that’s the max flow. but if it’s saturated I’m stuck.