
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
In the rectangular coordinate system on a plane, $O$ is the origin, vector $\overrightarrow{a}=(2,1)$, $A(1,0)$, $B(\cos \theta, t)$, and $\overrightarrow{a}$ is parallel to $\overrightarrow{AB}$.

(I) If $|\overrightarrow{AB}|=\frac{\sqrt{10}}{2}|\overrightarrow{OA}|$, find the coordinates of vector $\overrightarrow{OB}$.

(II) Find the maximum and minimum values of the function $y=\sin ^{2} \theta+\cos \theta-1$ with respect to $t^{2}$.

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