
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
Given the curve $C_1$ with the polar coordinate equation $\rho =1$, establish a rectangular coordinate system $xOy$ with the pole as the origin and the polar axis as the positive $x$-axis.

(1) If the curve $C_2: \begin{cases} x=1+t \\ y=2+t \end{cases}$ (with $t$ as the parameter) intersects the curve $C_1$ at two points $A,B$, find $| AB |$;

(2) If $M$ is a moving point on the curve $C_1$ with rectangular coordinates $(x,y)$, find the maximum value of $(x+1)(y+1)$.

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