>
>   Device-1: Broadcom Limited BCM4313 802.11bgn Wireless Network Adapter
>
ok, this is your card.

Lets see  output of
$ iw dev [your_dev_name]
it should be
 $ iw dev wlan0
I believe

Also, try to ping your wifi router ip address and some Internet address like
$ ping 8.8.8.8

Then, check https://www.speedtest.net/ : it gives a little bit more
information than "fast.com"

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