[Breaking News] Storm sweeping through a river!

In the image on the right, there is a river running between 2 banks (marked in green). There are 6 islands (marked in brown) in the river, and these land masses are connected by 13 bridges (marked in black). The inhabitants can only use these bridges to get across the river (no swimming, boats, etc. are allowed).

One day, an extremely heavy storm destroys at least one of the bridges, which prevents the inhabitants from crossing directly between the connected land masses. Out of the $$2^{13} - 1 = 8191$$ cases where at least one bridge is destroyed, in how many cases can the inhabitants still cross between the two banks?

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