
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 points $A(-2,0)$ and $B(2,0)$, the line $l$ passing through point $A$ and the line $m$ passing through point $B$ intersect at point $P$. Let the slope of line $l$ be $k_{1}$ and the slope of line $m$ be $k_{2}$, and ${k_1}{k_2}=-\frac{1}{4}$.  
$(1)$ Find the equation of the locus $C$ of point $P$;  
$(2)$ Given that the line $x=ty+1$ intersects the curve $C$ at points $M$ and $N$, and $O$ is the origin, if $\overrightarrow{OM}•\overrightarrow{ON}=0$, find the value of $t$.

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