
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 $\mathcal{C}$ be the hyperbola $y^2 - 4x^2 = 4$.  Given a point $P_0$ on the $y$-axis, we construct a sequence of points $(P_n)$ on the $y$-axis in the following manner: let $\ell_n$ be the line with slope -2 passing through $P_n$, then $P_{n+1}$ is the orthogonal projection of the point of intersection of $\ell_n$ and $\mathcal C$ onto the $y$-axis. (If $P_n = 0$, then the sequence terminates.)

Find the number of starting positions $P_0$ on the $y$-axis such that $P_0 = P_{1024}$. Your answer should use exponential notation, in simplest form.

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