7. Given the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$ with left and right foci $F_{1}, F_{2}$, a tangent line is drawn from $F_{1}$ to the circle $x^{2}+y^{2}=a^{2}$, intersecting the right branch of the hyperbola at point $P$, and $\angle F_{1} P F_{2}=45^{\circ}$. Then the eccentricity of the hyperbola is $\qquad$.
Given that $F_{1} P$ is tangent to circle $O$ at point $B$, and the midpoint of $F_{1} P$ is $C$, then Rt $\triangle O B C$ is an isosceles right triangle $\Rightarrow\left|F_{1} C\right|=b+a$, thus $\left|F_{1} P\right|=2(b+a) \Rightarrow\left|F_{2} P\right|=2 b \Rightarrow|O C|=b$. In $\triangle O F_{1} C$,
$$
c^{2}=(a+b)^{2}+b^{2}-\sqrt{2}(a+b) b \Rightarrow a=\sqrt{2} b \text {, }
$$

thus $c=\sqrt{3} a \Rightarrow e=\sqrt{3}$.