
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 oblique coordinate system $(xOy)$ with $\angle xOy = 45^\circ$, the oblique coordinates of point $P$ are defined as "$(x_0, y_0)$ if $\overrightarrow{OP} = x_0 e_1 + y_0 e_2$, where $e_1$ and $e_2$ are unit vectors in the same direction as the $x$-axis and $y$-axis of the oblique coordinate system, respectively." Given points $F_1(-1, 0)$ and $F_2(1, 0)$, find the equation of the locus of moving point $M(x, y)$ satisfying $|\overrightarrow{MF_1}| = |\overrightarrow{MF_2}|$ in the oblique coordinate system.

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