
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
As a puzzle game, Chinese chess has a long history. Behind Chinese chess lies the profound Chinese culture. In order to promote Chinese chess, a regional Chinese chess competition was held in a certain place, and Xiao Ming participated as a player. Among the other participants besides Xiao Ming, $50\%$ are Type 1 players, $25\%$ are Type 2 players, and the rest are Type 3 players. The probabilities of Xiao Ming winning against Type 1, Type 2, and Type 3 players are $0.3$, $0.4$, and $0.5$, respectively.  
$(1)$ Randomly select a player from the participants to compete against Xiao Ming. Find the probability of Xiao Ming winning.  
$(2)$ If Xiao Ming wins, find the probability that the player he competed against is a Type 1 player.

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