
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 $y^2 = 2px$ $(p > 0)$ with focus $F$ located on the line $x + y - 1 = 0$.
1. Find the equation of the parabola.
2. Draw a line passing through the focus $F$ at a 45-degree angle, intersecting the parabola at points $A$ and $B$. Find the horizontal coordinate of the midpoint $C$ of the line segment $AB$.

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