
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 a line $l$ that passes through point $P(1,1)$ with an angle of inclination $\alpha= \frac{\pi}{4}$, and a parabola $y^{2}=x+1$ intersects line $l$ at points $A$ and $B$. Find:
1. The value of $|PA| \cdot |PB|$
2. The length of chord $AB$
3. The coordinates of the midpoint $M$ of chord $AB$

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