
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
Let $P$ be a moving point on the circle $O: x^2 + y^2 = a^2$ $(a > 0)$, and let $PD \perp x$-axis at point $D$. Denote the locus of the midpoint $M$ of the line segment $PD$ as curve $C$.

(I) Find the equation of the curve $C$.
(II) If the moving line $l$ intersects the curve $C$ at points $A$ and $B$, and the area of $\triangle OAB$ (where $O$ is the origin) takes its maximum value, which is $1$, find the value of $a$.

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