Convert regex pattern to LL1 parser-CodePudding

Background: I'm trying to solve this leetcode problem: regular-expression-matching. My approach is to implement a LL(1) parser generator, might be overkilling for this problem but it's just a brain exercise.

I only have entry level knowledge of parser theory, so excuse me if this is a silly question that I ask.

So, there's this testcase that I fail. Requirement is regex pattern should match the whole string

Regex: a*a
Input: aaa

I cannot wrap my mind about how to convert this pattern into a LL(1) parser.

The way I see it, a*a pattern can be converted into production rules:

S -> Aa      # a*a
A -> aA | ε  # a*

Parse table:

	a	$
S	S -> Aa
A	A -> aA	A -> ε

Here's the steps to parse:

0: S$    aaa$  # use [S,a]
1: Aa$   aaa$  # use [A,a]
2: aAa$  aaa$  # eat 'a'
3: Aa$   aa$   # use [A,a]
4: aAa$  aa$   # eat 'a'
5: Aa$   a$    # use [A,a]
6: aAa$  a$    # eat 'a'
7: Aa$   $     # use [A,$]
8: a$    $     # Error!

The correct matching should be:

a* -> aa
a -> a

But what I get instead is:

a* -> aaa
a -> Error!

I don't know which part I'm missing