
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
In the diagram, the circle has radius  $\sqrt 7$  and and centre  $O.$  Points  $A, B$  and  $C$  are on the circle. If  $\angle BOC=120^\circ$  and  $AC = AB + 1,$  determine the length of  $AB.$ [asy]
import graph; size(120); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqttff = rgb(0,0.2,1); pen xdxdff = rgb(0.49,0.49,1); pen fftttt = rgb(1,0.2,0.2);
draw(circle((2.34,2.4),2.01),qqttff); draw((2.34,2.4)--(1.09,0.82),fftttt); draw((2.34,2.4)--(4.1,1.41),fftttt); draw((1.09,0.82)--(1.4,4.18),fftttt); draw((4.1,1.41)--(1.4,4.18),fftttt);
dot((2.34,2.4),ds); label(" $O$ ", (2.1,2.66),NE*lsf); dot((1.09,0.82),ds); label(" $B$ ", (0.86,0.46),NE*lsf); dot((4.1,1.41),ds); label(" $C$ ", (4.2,1.08),NE*lsf); dot((1.4,4.18),ds); label(" $A$ ", (1.22,4.48),NE*lsf); clip((-4.34,-10.94)--(-4.34,6.3)--(16.14,6.3)--(16.14,-10.94)--cycle);
[/asy]

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