
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
Let  $\mathcal E$  be an ellipse with foci  $F_1$  and  $F_2$ .  Parabola  $\mathcal P$ , having vertex  $F_1$  and focus  $F_2$ , intersects  $\mathcal E$  at two points  $X$  and  $Y$ .  Suppose the tangents to  $\mathcal E$  at  $X$  and  $Y$  intersect on the directrix of  $\mathcal P$ .  Compute the eccentricity of  $\mathcal E$ .

(A *parabola*  $\mathcal P$  is the set of points which are equidistant from a point, called the *focus* of  $\mathcal P$ , and a line, called the *directrix* of  $\mathcal P$ .  An *ellipse*  $\mathcal E$  is the set of points  $P$  such that the sum  $PF_1 + PF_2$  is some constant  $d$ , where  $F_1$  and  $F_2$  are the *foci* of  $\mathcal E$ .  The *eccentricity* of  $\mathcal E$  is defined to be the ratio  $F_1F_2/d$ .)

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