
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 $xOy$, the points $A(-1,-2), B(2,3), C(-2,-1)$ are given.

$(1)$ Calculate the lengths of the diagonals of the parallelogram with sides $AB$ and $AC$.

$(2)$ Let the real number $t$ satisfy $(\vec{AB}-t\vec{OC}) \cdot \vec{OC}=0$, find the value of $t$.

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