
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 points $F\_1(-1,0)$ and $F\_2(1,0)$, the sum of the distances from moving point $P$ to points $F\_1$ and $F\_2$ is $4$.

1. Determine the shape of point $P$'s trajectory ($C$) and write its equation.
2. If curve $C$ intersects with line $m$: $y=x-1$ at points $A$ and $B$, find the length of chord $AB$.

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