
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 \(y^{2}-x^{2}=1\). Given a point \(P_{0}\) on the \(x\)-axis, we construct a sequence of points \((P_{n})\) on the \(x\)-axis in the following manner: let \(\ell_{n}\) be the line with slope 1 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 \(x\)-axis. (If \(P_{n}=0\), then the sequence simply terminates.)

Let \(N\) be the number of starting positions \(P_{0}\) on the \(x\)-axis such that \(P_{0}=P_{2008}\). Determine the remainder of \(N\) when divided by 2008.

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