Washington-(CNN)In the buildup to the current Ukrainian counteroffensive, the US urged Kyiv to keep the operation limited in both its objectives and its geography to avoid getting overextended and bogged down on multiple fronts, multiple US and western officials and Ukrainian sources tell CNN.

Those discussions involved engaging in "war-gaming" with Kyiv, the sources said -- analytical exercises that were intended to help the Ukrainian forces understand what force levels they would need to muster to be successful in different scenarios.

The Ukrainians were initially considering a broader counteroffensive, but narrowed their mission to the south, in the Kherson region, in recent weeks, US and Ukrainian officials said.

Pentagon spokesperson Brig. Gen. Pat Ryder told CNN that "the United States has routine military-to-military dialogue at multiple levels with Ukraine. We will not comment on the specifics of those engagements. Generally speaking, we provide the Ukrainians with information to help them better understand the threats they face and defend their country against Russian aggression. Ultimately, the Ukrainians are making the final decisions for their operations."

Officials say they believe there is now increased parity between the Ukrainian and Russian militaries. But western officials have been hesitant to label the nascent Ukrainian operation -- which appeared to begin on Monday in the southern province of Kherson -- a true "counteroffensive."

How successful Ukraine is likely to be in regaining lost territory remains an open question, sources familiar with the latest intelligence tell CNN. Ukrainian officials have already said this offensive will likely be a slow operation, and punishingly cold winter weather is coming and then an early spring mud, both of which could force pauses in the fighting.

Still, there is a distinct feeling amongst Ukraine's US and western advisers that the Ukrainian military is on much more even footing with Russia than was believed even just a few short months ago, multiple officials told CNN. Russia still maintains superior numbers in overall manpower and massed artillery.

But Ukrainian capabilities, bolstered by sophisticated western arms and training, have closed an important gap, officials say -- particularly the High Mobility Artillery Rocket Systems, or HIMARS, that Ukraine has been using to launch attacks behind Russian front lines in recent months.

"It shows you what the sustained training and weapons provision can do when the force is highly motivated and capable in its employment," a senior NATO official told CNN.

Another US military source put it more bluntly: Ukraine has made up for Russia's advantage in sheer volume of fire with its "competence."