
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 polar coordinate system, let curve $C_{1}$ be described by $\rho = 2\sin \theta$ and curve $C_{2}$ by $\rho \cos \theta = 3$. With the pole as the origin and the polar axis as the positive half of the $x$-axis, establish the Cartesian coordinate system.
$(1)$ Find the Cartesian equations for curves $C_{1}$ and $C_{2}$.
$(2)$ Let line $l$ pass through point $P(1,\pi)$ and intersect $C_{1}$ at points $A$ and $B$, and $C_{2}$ at point $Q$. If $|PA| + |PB| = \lambda |PQ|$, find the maximum value of $\lambda$.

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