
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 that $O$ is the coordinate origin, let $({F_1},{F_2})$ be the left and right foci of the hyperbola $({x^2} - {y^2} = 1)$, and let $P$ be any point on the hyperbola. Find the distance from $O$ to the foot of the perpendicular line drawn from the left focus ${F_1}$ to the bisector of $\angle {F_1}P{F_2}$.

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