
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 an ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with foci $F_{1}$ and $F_{2}$, where $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse, respectively. The parabola $E: y^2 = 4x$ shares the same foci with the ellipse $C$. Point $P$ is the intersection point of the parabola $E$ and the ellipse $C$ in the first quadrant. The line $PF_{1}$ is tangent to the parabola $E$. Calculate the length of the major axis of the ellipse $C$.

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