
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In a chess competition, the order of moves is determined by flipping poker cards to compare their values. There are five poker cards with values of $1$, $2$, $3$, $4$, and $5$. The backs of the cards are indistinguishable. The cards are placed face down, and one player flips over a card first, then the other player flips over one of the remaining four cards. The player with the higher value goes first.  
$(1)$ If player A flips over a card with a value of $4$ first, the probability that player A goes first is ______.  
$(2)$ In two rounds of the game, assuming player A flips over cards with a value of $3$ both times, find the probability that player A goes first in both rounds (solve using a tree diagram or list method).

Present the answer in LaTex format: \boxed{Your answer}
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