
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 the function $f(x)=\sin (\omega x+\varphi)$ $(\omega > 0, 0 < \varphi < \frac{\pi}{2})$ whose graph passes through the point $(0, \frac{1}{2})$, and the distance between two adjacent axes of symmetry is $\frac{\pi}{2}$.  
Ⅰ Find the expression of the function $f(x)$ and its intervals of monotonic increase;  
Ⅱ In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. If $f\left( \frac{A}{2}\right)-\cos A= \frac{1}{2}$ and $bc=1$, $b+c=3$, find the value of $a$.

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