
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 curve $C_1$: $y^2 = tx$ ($y > 0$, $t > 0$) has a tangent at point $M\left(\frac{4}{t}, 2\right)$ that is also tangent to the curve $C_2$: $y = e^{x+1} - 1$, find the value of $t$.

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