As shown in the figure, in an $8 \times 8$ square grid, there are two bugs at points $A$ and $B$ (the bug at point $A$ is called bug $a$, and the bug at point $B$ is called bug $b$). Each bug moves 1 grid (choosing one of the four directions: up, down, left, or right). If bug $b$ moves two times, and bug $a$ moves three times, the number of ways in which the distance from bug $b$ to point $A$ is less than or equal to the distance from bug $a$ to point $A$ is $\qquad$ ways.
Counting, Multiplication Principle, Addition Principle, Permutations, Combinations
(1) $b$ bug moves 2 steps
$\rightarrow \rightarrow$, finally $b$ bug is 3 steps away from point $A$
$\leftarrow \leftarrow$, finally $b$ bug is 1 step away from point $A$
$\uparrow \uparrow$ or $\downarrow \downarrow$, finally $b$ bug is $\sqrt{5}$ steps away from point $A$ (diagonal of a $2 \times 1$ rectangle)
$\rightarrow \uparrow$ or $\rightarrow \downarrow$ and their permutations, finally $b$ bug is 1 step away from point $A$
$\leftarrow \uparrow$ or $\leftarrow \downarrow$ and their permutations, finally $b$ bug is 1 step away from point $A$
$\uparrow \downarrow$ and its permutations, finally $b$ bug is 1 step away from point $A$

Finally, $b$ bug is 1 step away from point $A$, there are $1+4 \times P_{2}^{2}=9$ ways
Finally, $b$ bug is $\sqrt{5}$ steps away from point $A$ (diagonal of a $2 \times 1$ rectangle), there are $2+2 \times P_{2}^{2}=6$ ways
Finally, $b$ bug is 3 steps away from point $A$, there is 1 way
(2) $a$ bug moves 3 steps
$\rightarrow \rightarrow \rightarrow$ or $\uparrow \uparrow \uparrow$ or $\leftarrow \leftarrow \leftarrow$ or $\downarrow \downarrow \downarrow$, finally $a$ bug is 3 steps away from point $A$
$\rightarrow \rightarrow \leftarrow$ or $\leftarrow \leftarrow \rightarrow$ or $\uparrow \uparrow \downarrow$ or $\downarrow \downarrow \uparrow$ and their permutations, finally $a$ bug is 1 step away from point $A$
$\rightarrow \uparrow$ or $\rightarrow \rightarrow \downarrow$ or $\leftarrow \uparrow$ or $\leftarrow \downarrow \downarrow$ or $\uparrow \uparrow \rightarrow$ or $\uparrow \uparrow \leftarrow$ or $\downarrow \downarrow \rightarrow$ or $\downarrow \downarrow \leftarrow$ and their permutations
Finally, $a$ bug is $\sqrt{5}$ steps away from point $A$ (diagonal of a $2 \times 1$ rectangle)
$\rightarrow \uparrow \leftarrow$ or $\rightarrow \uparrow \downarrow$ or $\rightarrow \leftarrow \downarrow$ or $\uparrow \leftarrow \downarrow$ and their permutations, finally $a$ bug is 1 step away from point $A$
Finally, $a$ bug is 1 step away from point $A$, there are $4 \times C_{3}^{1} \times C_{2}^{2}+4 \times P_{3}^{3}=36$ ways
Finally, $a$ bug is $\sqrt{5}$ steps away from point $A$ (diagonal of a $2 \times 1$ rectangle), there are $8 \times C_{3}^{1} \times C_{2}^{2}=24$ ways
Finally, $a$ bug is 3 steps away from point $A$, there are 4 ways
(3) Finally, the distance between $b$ bug and point $A$ is less than or equal to the distance between $a$ bug and point $A$
(4) When finally $b$ bug is 1 step away from point $A$
Then finally $a$ bug is 1 step or $\sqrt{5}$ steps (diagonal of a $2 \times 1$ rectangle) or 3 steps away from point $A$
There are $9 \times (36+24+4)=9 \times 64=576$ ways
(2) When finally $b$ bug is $\sqrt{5}$ steps away from point $A$ (diagonal of a $2 \times 1$ rectangle)
Then finally $a$ bug is $\sqrt{5}$ steps (diagonal of a $2 \times 1$ rectangle) or 3 steps away from point $A$
There are $6 \times (24+4)=168$ ways
(3) When finally $b$ bug is 3 steps away from point $A$
Then finally $a$ bug is 3 steps away from point $A`
There are $1 \times 4=4$ ways
In summary, the number of ways for the distance between $b$ bug and point $A$ to be less than or equal to the distance between $a$ bug and point $A$ is $576+168+4=748$ ways