
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
In the Cartesian coordinate system $xOy$, with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The parametric equation of the curve $C$ is 
$$
\begin{cases} 
x = 2 \cos \alpha + 1 \\
y = 2 \sin \alpha 
\end{cases}
(\alpha \text{ is the parameter}).
$$
The polar equation of the line $l$ is $\rho \sin (\theta + \frac{\pi}{4}) = \sqrt{2}$. The line $l$ intersects curve $C$ at points $M$ and $N$. Find the length of $MN$.

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