
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 a point M(1, a).
1. If there is exactly one line through point M that is tangent to circle O, find the value of the positive number a, and derive the equation of the tangent line.
2. If $a=\sqrt{2}$, and two chords AC and BD of the circle passing through point M are perpendicular to each other:
   i. Find the maximum area of quadrilateral ABCD.
   ii. Find the maximum value of $|AC| + |BD|$.

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