
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 rectangular coordinate system on a plane, the parametric equation of line $l$ is given by $\begin{cases} x=1+t \\ y=2+t \end{cases}$ (where $t$ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the non-negative semi-axis of the $x$-axis as the polar axis. The polar equation of circle $C$ is $\rho^2 + 2\rho \sin \theta = 3$. If line $l$ intersects circle $C$ at points $M$ and $N$, find the length of the line segment $MN$.

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