
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 an ellipse $E$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, $F\_1(-c,0)$, $F\_2(c,0)$ are the two foci of the ellipse, $M$ is any point on the ellipse, and $|MF\_1|$, $|F\_1F\_2|$, $|MF\_2|$ form an arithmetic sequence, and the length of the chord passing through the foci of the ellipse and perpendicular to the major axis is $3$.
(1) Find the equation of the ellipse $E$;
(2) If there exists a circle with the origin as the center, such that any tangent line to this circle intersects the ellipse $E$ at two points $A$ and $B$, and $\overrightarrow{OA} \perp \overrightarrow{OB}$, find the equation of this circle.

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