
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 $\mathscr{P}$ be the parabola in the plane given by the equation \( y = x^2 \). Let \(\Gamma_1\) be the circle with diameter 1 that is internally tangent to \(\mathscr{P}\) at the origin. By recurrence, \(\Gamma_{n+1}\) is defined as the circle tangent to \(\Gamma_n\) and twice to \(\mathscr{P}\). Calculate the diameter of \(\Gamma_{2007}\).

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