
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 functions $f(x)=x^{2}+c$ and $g(x)=ae^{x}$ whose graphs have a common point $P(2,t)$, and the curves $y=f(x)$ and $y=g(x)$ have the same tangent line at point $P$. If the negative zero of the function $f(x)-g(x)$ lies in the interval $(k,k+1) (k\in \mathbb{Z})$, then $k=$ \_\_\_\_\_\_.

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