
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
The parabola $M\_1$: $y=ax^{2}+c$ is translated to obtain the parabola $M\_2$. The parabola $M\_2$ passes through the vertex $A$ of the parabola $M\_1$. The axis of symmetry of the parabola $M\_2$ intersects the parabolas $M\_1$ and $M\_2$ at points $B$ and $C$, respectively. If the coordinates of point $C$ are $(2, c-5)$, find the area of $\triangle ABC$.

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