
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
Let $\mathcal{D}$ be the hyperbola $y^2 - 2x^2 = 2$. Suppose we are given a point $P_0$ on the $x$-axis, and we construct a sequence of points $(P_n)$ on the $x$-axis based on the following iterative process: Let $\ell_n$ be the line with a slope of 2 passing through $P_n$, and let $P_{n+1}$ be the orthogonal projection of the point of intersection between $\ell_n$ and $\mathcal{D}$ onto the $x$-axis. The sequence terminates if $P_n$ coincides with the origin.

Determine the number of initial positions $P_0$ on the $x$-axis such that $P_0 = P_{2048}$. The answer should be provided using exponential notation, in simplest form.

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