
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 circle, $M$ is the midpoint of arc $CAB$ and segment $MP$ is perpendicular to chord $AB$ at the midpoint $P$. If the length of chord $AC$ is $x$ and the length of segment $AP$ is $2x + 2$, find the length of segment $PB$.

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