
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
The parametric equations for curve \( C_1 \) are
$$
\begin{cases}
x=8t^{2} \\
y=8t
\end{cases}
$$
where \( t \) is the parameter. In a polar coordinate system (using the same unit length as the Cartesian coordinate system xy, with the origin O as the pole, and the positive x-axis as the polar axis), the polar equation for circle \( C_2 \) is \( \rho = r \) (where \( r > 0 \)). If a line with slope 1 passes through the focus of \( C_1 \) and is tangent to \( C_2 \), then \( r = \) ______ .

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