
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 the curve $E$: $\frac{x^2}{a^2} + y^2 = 1$ ($a > b$, $a \neq 1$) and two points $A(x_1, y_1)$, $B(x_2, y_2)$ ($x_1 \neq x_2$) on it.

$(1)$ If points $A$ and $B$ are both on the line $y = 2x + 1$, and the midpoint of segment $AB$ has an x-coordinate of $- \frac{1}{3}$, find the value of $a$;

$(2)$ Let $\overrightarrow{m} = (\frac{x_1}{a}, y_1)$, $\overrightarrow{n} = (\frac{x_2}{a}, y_2)$. If $\overrightarrow{m} \perp \overrightarrow{n}$ at the origin, investigate whether the area of $\triangle OAB$ is a constant value. If yes, find the constant value; if not, explain why.

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