
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 a moving point $P(x,y)$ and two fixed points $A(-2,0), B(2,0)$ on a plane, the slopes of the lines $PA$ and $PB$ are ${k_1}$ and ${k_2}$ respectively, with ${k_1} \cdot {k_2} = -\frac{1}{4}$.

(1) Find the equation of the trajectory $C$ of the moving point $P$.

(2) Let line $l: y=2x+m$ intersect curve $C$ at two distinct points $M$ and $N$ such that $OM$ is perpendicular to $ON$ (with $O$ being the origin). Find the value of $m$.

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