
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 the right vertex of the hyperbola $\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1$ be $A$, and the right focus be $F(c, 0)$. A chord $PQ$ passes through $F$ and is perpendicular to the x-axis. Perpendicular lines are drawn from points $P$ and $Q$ to line $AQ$ and $AP$ respectively, and these perpendicular lines intersect at point $B$. If the distance from $B$ to line $PQ$ is less than $2(a+c)$, determine the range of the eccentricity of this hyperbola.

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