
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 parabola $y^2=2px$, and a line with an inclination angle of $\frac{\pi}{4}$ intersects the parabola at points $A$ and $B$. If the midpoint of chord $AB$ is $Q(3,2)$,

$(1)$ Find the equation of the parabola.

$(2)$ Find the length of the chord $|AB|$.

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