
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
Given that $P$ is a point inside circle $O$ such that it is equidistant from four points $A, B, C, D$ on the circle, forming chords $PA, PB, PC, PD$ with lengths $3, x, 5, 4$ respectively, determine the value of $x$ in inches.

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