
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 hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \left( a > 0, b > 0 \right)$ has an eccentricity of $\sqrt{2}$, and a tangent line is drawn from the left focus $F_1(-c,0)$ to the circle $x^2 + y^2 = a^2$ at point $E$. Extend $F_1E$ to intersect the parabola $y^2 = 4cx$ at point $P$. Calculate the length of the segment $PE$.

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