
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 the arc $CAB$ and the segment $MP$ is perpendicular to the chord $AB$ at $P$. The circle has a second chord $CC'$ such that $C$ is between $A$ and $C'$ and $AC + CC' = AB$. If the length of chord $AC$ is $2x$ and segment $AP$ is $(2x+2)$, find the length of segment $PB$.
A) $3x + 4$
B) $2x + 3$
C) $4x + 4$
D) $2x + 2$
E) $4x$

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