
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 the circle $O: x^2 + y^2 = 4$ and the point $M(1, a)$.
(1) If there is exactly one line passing through point $M$ that is tangent to circle $O$, find the value of the real number $a$ and the equation of the tangent line;
(2) If $a = \sqrt{2}$, find the equations of the lines containing the shortest chord $AC$ and the longest chord $BD$ passing through point $M$. Also, find the area of quadrilateral $ABCD$ at this time.

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