A $k$-input binary truth table is a map from $k$ input bits (binary digits, $0$ [false] or $1$ [true]) to $1$ output bit. For example, the $2$-input binary truth tables for the logical $\text{AND}$ and $\text{XOR}$ functions are:

$x$	$y$	$x\text{AND}y$
$0$	$0$	$0$
$0$	$1$	$0$
$1$	$0$	$0$
$1$	$1$	$1$

$x$	$y$	$x\text{XOR}y$
$0$	$0$	$0$
$0$	$1$	$1$
$1$	$0$	$1$
$1$	$1$	$0$

How many $6$-input binary truth tables, $\tau$, satisfy the formula $$\tau (a,b,c,d,e,f)\text{AND}\tau (b,c,d,e,f,a\text{XOR}(b\text{AND}c))=0$$ for all $6$-bit inputs $(a,b,c,d,e,f)$?