
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 Cartesian coordinate system $Oxy$, where $O$ is the origin, point $A(0,4)$, $B(0,2)$, the plane vectors $\overrightarrow{OA}$, $\overrightarrow{OB}$ satisfy $(2\overrightarrow{OC}-\overrightarrow{OA})(\overrightarrow{OC}-\overrightarrow{OB})=0$. For any real number $t < 0$ and any vector $\overrightarrow{OC}$ that meets the condition, the minimum value of $|\overrightarrow{OC}-\frac{1}{4}t\cdot\overrightarrow{OA}-\frac{1}{2}[\ln(-t)-1]\cdot\overrightarrow{OB}|$ is \_\_\_\_\_\_.

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