
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 parabola $y=a\left(x+2\right)^{2}+3$ is shifted $2$ units to the right and then $3$ units down, passing through the point $M\left(1,1\right)$.<br/>$(1)$ Find the analytical expression of the parabola after the translation;<br/>$(2)$ Point $A$ lies on the translated parabola, to the right of the axis of symmetry. Point $A$ has coordinates $(t, t^{2})$. Point $B$ is obtained by rotating point $A$ counterclockwise $90^{\circ}$ around the origin. If line $l$<span class="special-symbol">∥</span>$OB$ and $l$ intersects the translated parabola at only one point $C$, find the analytical expression of line $AC$ when the distance from point $D(\frac{1}{2},0)$ to line $AC$ is maximized.

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