
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 circle $O$, point $M$ is the midpoint of arc $CAB$, and segment $MP$ is perpendicular to chord $AB$ at $P$. If the measure of chord $AC$ is $2x$ and that of segment $AP$ is $(3x+2)$, then determine the measure of segment $PB$.
**A)** $3x + 1$  
**B)** $3x + 2$  
**C)** $5x + 4$  
**D)** $6x + 4$  
**E)** $4x + 2$

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