
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
Select a point $X$ inside or on the circumference of a circle. Through point $X$, draw two mutually perpendicular chords $AC$ and $BD$ (when $X$ is on the circumference, the degenerate cases are allowed, i.e., one chord is the diameter, and the other chord shrinks to a point). For all possible choices of $X$, find the maximum and minimum values of the sum $S=|AC| + |BD|$.

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