
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 $xOy$, there are three points $A(a,1)$, $B(2,b)$, and $C(3,4)$. If the projections of $\vec{OA}$ and $\vec{OB}$ onto the direction of $\vec{OC}$ are the same, then find the minimum value of $a^{2}+b^{2}$.

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